Tropes used in New Avengers Volumes 1 & 2 include:
A Day in the Spotlight: During Civil War, each issue focused on an individual member of the team who didn't have their own series, with Captain America and Iron Man also getting issues due to scheduling conflicts with their own series.
Back for the Finale: Wolverine, after being unavailable for the last few arcs due to events in X-Men.
Badass in Distress: The first things the second line-up do are try to retrieve Captain America's body from S.H.I.E.L.D. after learning that he may still be alive, before flying to Japan to save Echo from being forced into the Hand.
Book Ends: Vol. 2 #16 (The final Fear Itself tie-in) opened with Hawkeye questioning what makes someone "Avengers material" and several Avengers arguing that certain people don't belong on the team; the issue ends with the same Avengers pointing out that applying Alternate Character Interpretation means that same person is very much the type of person who should be an Avenger.
In a wider sense, Vol. 2 started and ended with arcs heavily based around magic, possession, Doctor Strange, and the title of Sorcerer Supreme.
Brick Joke: In the first issue of Vol. 2, Iron Fist gave Luke Cage a dollar to purchase Avengers Mansion from Tony Stark. When Luke leaves the team in the final issue of the volume, he sells it back to Stark for 5 dollars and Iron Fist complains that it was his money in the first place, whilst Stark notes that he sold the mansion to Cage for a fifth of that.
Iron Fist(mid-battle): Awww... is this one of those 'witty banter' teams?
Iron Fist: I hate banter.
Luke Cage: You're doing banter.
Chekhov's Gunman: Ronin. Despite being on all of the early covers, the character didn't appear until the midway point of the original line-up's run (and then immediately went to Japan to spy on the Hand), before becoming part of the team properly after Civil War.
Chuck Cunningham Syndrome: Echo, after Secret Invasion. Lampshaded by Spider-Man asking if anyone knew where she'd gone; and again with her reappearance when Luke Cage and Jessica Jones were looking for a Nanny, as she angrily asks if Cage even remembers that she used to be on the team.
Establishing Character Moment: The Sentry grabs Carnage and flies him out into space and tears the symbiote in two. The second arc established the Sentry's character to a fuller extent, but this established that he was not going to be a low-level hero.
Everyone Went to School Together: Spider-Man and Jessica Jones, complete with the embarrassing revelations that she had a crush on him; and that Peter didn't even realise that Jessica Jones, superhero and former Bugle co-worker, went to the same school as him as he only knew her as "Coma Girl".
Hidden in Plain Sight: When the team were hiding out at the Sanctum Sanctorum, Doctor Strange cast a spell on the building that it made it appear as if it had been long been deserted. He even went so far as to use a codeword which also hid any inhabitants that used it. Subverted, as Iron Man suspected the team were doing this, but couldn't find a magic user (or at least, a magic user willing to work with S.H.I.E.L.D.) able to overcome Strange's spell.
It's Not You, It's My Enemies: Daredevil pulls this when Captain America first approaches him to be part of the first line-up. He tries to pull it again when Luke Cage and Jessica Jones ask him to join the team following Fear Itself, but they point out it doesn't really work, since he'd be an Avenger and they'd have his back.
Doctor Strange tries a similar argument after Daniel Drumm swears vengeance on him, which sets up one of Luke Cage's better lines. "Okay. Raise your hand if any of you [the gathered members of the New Avengers] have ever had the bad guy threaten all holy hell on you before he died, went to jail, disappeared, vanished, or just in general felt like lipping off." Every hand in the room goes up.
On the other hand, Jessica Jones walked away from the team because her daughter's life was threatened by Norman Osborn.
Subverted - After the death of Steve Rogers, Luke Cage states that they are Avengers, because Captain America said they were.
Played straight, however, in Clint Barton assuming the Ronin name after Captain America's death and keeping it until the start of the Heroic Age; and Bucky joined the team during his tenure as Captain America.
Discussed by Iron Fist and Doctor Strange following Civil War, with the two of of them bearing mantles (Sorcerer Supreme and the Iron Fist) that had been previously passed down to them, and they will later pass on in their lives.
Loophole Abuse: In the initial arc, Maria Hill informs Captain America that he can't just form a new team of Avengers because he feels like it. Cap then tells her that she can check his S.H.I.E.L.D. file and verify for herself that he was previously given "Full Champion License" during Nick Fury's tenure as S.H.I.E.L.D. director; which means that he has the authority to assemble any team he sees fit to go on any mission he sees fit and they don't need S.H.I.E.L.D. permission to put the Avengers back together.
When Brother Voodoo challenges Agamotto one-on-one, he uses a spell that gathers the collective strength and fighting ability of the New Avengers into a single person.
Similarily, Spider-Man reminds everyone that Victoria Hand was Norman Osborn's right hand during Dark Reign at every opportunity.
New Powers as the Plot Demands: The Hood was granted the Norn Stones by Loki after his connection with Dormammus was severed. After Loki took them back, he sought out Count Nefaria to try and invoke this again, but the New Avengers tracked him down and arrested him before he could get new powers again.
One of the Boys: Within the team dynamic, Mockingbird tends to be this. She's not part of the Romantic Three Girl Friendship between Carol and the two Jessicas, and instead whenever she's getting chummy with a teammember, it's with one of the boys. She also has the most tomboyish personality, probably helped by her tomboyish name.
Jessica Drew/Spider Woman was revealed to be recruited by HYDRA to infiltrate S.H.I.E.L.D. as The Mole in exchange for her powers being restored, except Nick Fury had her feeding HYDRA limited intel whilst S.H.I.E.L.D. analysed the HYDRA cell and planned to use the intel to take them down; when the Captain America recruited her to the New Avengers, HYDRA amended her role to also spy on the Avengers. When the team found out about her role as a double agent, it was decided to do the same thing she was doing for S.H.I.E.L.D. and have her feed HYDRA limited intel. And then it turns out that she was replaced by Veranke during the proceedure to restore her powers, and she was playing the New Avengers, HYDRA and S.H.I.E.L.D. as part of the Skrull's plans to take Earth.
Victoria Hand, formerly Norman Osborn's right hand during Dark Reign, serves as the team's S.H.I.E.L.D liasion at Steve Rogers's request, but provides info on the team to Osborn's new H.A.M.M.E.R organisation. However, this is also at Rogers's request, as she provides info on H.A.M.M.E.R to him, with the New Avengers' distrust of her providing the ideal cover.
Put on a Bus: Wolverine, for the last few arcs of the first volume, due to his being on Utopia with the rest of the X-Men. Notable, as he was still considered a member of the team during this time - Everyone else who left the title wasn't.
Ragtag Bunch of Misfits: By vol. 2 under Luke Cage, the team consists of anyone he has managed to recruit, yielding a mismatched bag of heroes, some of whom are on their second (or third, or fourth) chance. Wong called them a "second-rate pile of Avengers" and Kyle Richmond, aka Nighthawk, said "clearly you guys are The Defenders."
Terrible Interviewees Montage: Luke and Jessica trying to find a superpowered nanny for their daughter. Some candidates are horrible, some are offended, and some take the opportunity to get in a few zings at the team. But the last one is perfect.
Wolverine: [to Brother Voodoo] Bub, I can do what needs to be done in a way you can't.
The Thing: I think that's a compliment, by the way.
Trauma Conga Line: This series has been hard on Doctor Strange. The final flare-up of dark magic which led him to give up the mantle of Sorcerer Supreme; the quest for his successor with a Dormammu-empowered Hood at his heels; and finally the abrupt invasion of Agamotto into their dimension, which the new Sorcerer Supreme, Brother Voodoo, repelled at the cost of the Eye of Agamotto and his own life — all happened in issues of New Avengers.
What Happened to the Mouse?: The first time Earth has no Sorcerer Supreme, it's an immediate crisis that must be dealt with at once. The second time, it appears that two years of real time can pass without any particular badness occurring.
Wolverine Publicity: There are three characters who were in the book since the very first arc - Luke Cage, Spider-Man, and Wolverine. Guess which one's the Creator's Pet and which two this trope applies to.
Tropes used in New Avengers Volumes 3 include:
Another Story for Another Time: Tony gives this explanation to Reed when discussing what he was doing in space during the time skip between issues while remembering meeting the Watcher on the moon.
Apocalypse How: Every incursion means that Class X (planetary destruction) is the least horrific thing that could happen. An incursion that goes on to its fatal result means Class X-5 (multiversal destruction) will happen, and the whole system seems to be marching with horrible inevitability toward Class Z (the end of everything).
How we see that Captain America distantly remembers being mind-wiped.
Black Swan weeps and cries for help in her sleep, showing that she still has a heart.
Bilingual Bonus: Black Swan peppers her speech with Sumerian. Most of it is translated, but not all.
Canon Immigrant: The alternate version of Terrax from one of the Counter-Earths, who is defeated and imprisoned into Earth-616 by the Illuminati.
Cast from Hit Points: Most of the initial plans the Illuminati have in issue 4 for taking out another world in a incursion event have this drawback.
Comm Links: Each of the members has a device hidden within his palm to confirm their identities and for short communications with one another in times of crisis.
Cross Through: Hickman currently writes both Avengers and New Avengers, and definitely has plans for them interacting. For example, he said that the themes for the comics is that Avengers is about life, and New Avengers, death. Also, in the first issue of Avengers, Cap has a dream based on the discussions with the Illuminati. The end of that comic's first arc seems to imply that Cap and Iron Man will eventually butt heads since Iron Man is on both teams and gave the order to mind-wipe Cap in the first place.
Cycle of Revenge: The Cold War between Wakanda and Atlantis which after the first arc escalates into full war with attacks from both camps. Even after Namor with T'Challa tried to broker a peace in secret, T'Challa's sister and current Queen Shuri has Wakanda launch brutal attack on Atlantis just before the Infinity tie in starts. Namor after being confronted in the rubble of his city by Proxima Midnight and her forces surrenders and when questioned about the location of the last Infinity Gem tells her it is in Wakanda.
Dirty Business: Incursions force people to choose to destroy another Earth to spare their own. The first few are resolved for the Illuminati by others, but Black Swan repeatedly warns them that they won't be able to keep their hands clean forever.
Disaster Dominoes: One universe's early destruction has set off a chain of universes collapsing into each other, threatening to destroy the entire multiverse.
Enemy Mine: Namor and T'Challa. Since the events of ''Avengers Vs X-Men', they rule kingdoms that are at war with each other. As soon as he calls the team in, T'Challa takes some time out to inform Namor that when the current crisis is over, he plans to kill him.
Expendable Alternate Universe: The heroes really don't want to play it straight, since they're aware that, even if the other world isn't theirs, it still means murdering every single person on it.
A few issues actually have the Illuminati watching different alternate worlds meet their end at the hands of threats they themselves have been fortunate to not yet challenge directly.
Godzilla Threshold: The events of the first issue, being unable to save the teens with him and watching a world be destroyed cause Black Panther to assemble the Illuminati.
When the Infinity Gauntlet breaks, the Illuminati realize they must be willing to at least consider destroying entire worlds to save the rest. The lone dissenter to this is expelled.
They ultimately realize they could not condemn billions to death no matter the stakes, when confronted with an alternate Galactus that was willing to do what they had spent the entire issue planning on doing and destroy the alternate Earth that would collide with theirs to save the rest of his universe. However, they did let him have his world when it was clear there was no time left to save it.
The Heart: Captain America. His determination to stay this in the face of universal destruction is why they kick him out.
Hope Spot: Issue 3 The group reassembles the Infinity Gauntlet in order to push oncoming universe's Earths away from their own. With Cap wielding it the plan appears to work... but the Infinity Gems are destroyed, leaving the Illuminati without their greatest weapon and no way to stop future incursions.
Jigsaw Puzzle Plot: The biggest problem for the heroes is that they have no idea why the incursions are happening, or how to stop them without resorting to destroying the other earth. Luckily, Black Swan has some answers.
Nice Job Breaking It, Hero: The Illuminati bring the Infinity Gems out of hiding in the hope that the Infinity Gauntlet can resolve incursions without the need to destroy the other Earth. It works... once... and then breaks, but not before alerting Galactus, the Watcher, and Thanos to its presence.