Most times Mugen goes to rescue Fuu. One that stands out in particular is episode 4 ("Hellhounds for Hire (Part 2)"), when Mugen stops at the crossroads where he and Jin ditched Fuu. Mugen stands there in the rain as she flashes in his mind - "WHAT ABOUT YOUR PROMISE NOT TO KILL EACH OTHER?!" - swears violently, and goes back not to fight Jin, but to save Fuu. That moment when he kicks his way into the brothel, his sword drenched in blood, and demands, "Where's the girl?!" - it's like an extreme example of Aww, Look! They Really Do Love Each Other.
Jin saving Shino from her unwanted brothel life which she had reluctantly accepted.
Episode 15 ("Bogus Booty"), when Mugen is alone with a female ninja posing as a whore. Realising her partner is in danger she tries to knock him out with a vase, finds him incredibly tough and instead manipulates him with sexual promises into single-handedly taking down a crime syndicate who were gearing up to torture her partner to death. At the end, when they are laying an unconscious Mugen underneath a tree to recover, the partner asks who he is and she simply replies "That is the man I am going to marry" before leaving and voicing the hope that he finds her again when he is ready.
In Episode 18 ("War of Words"), when confronted with why Mugen would tag his companions and their belongings with graffiti, Mugen replies offhandedly, "You gotta write your name on your stuff, you know?" Truthfully, it's more like he labels them (writing "Fuu" on Fuu's clothing, and "animal" on Momo's stomach), but the tagging implies that he regards Jin, Fuu, and Momo with some sentimental value, even as "belongings".
A quiet but huge moment in Episode 19 ("Unholy Union") at Xavier's church - Mugen draws his sword and heads for Xavier, but Jin lays a hand on his shoulder to stop him, and Mugen - a crazy stubborn brawler who takes orders from NO one - shoots him a look, sees that Jin's reading the situation carefully, and steps back. A perfect illustration of the (however grudging) respect and understanding that have developed between them.
The final three-episode arc of the series, "Evanescent Encounter", which is chock full of crowning moments of all kinds:
Fuu is feeling melancholy because their journey is coming to an end. She leans into him and Jin puts a comforting hand on her shoulder.
When both Jin and Mugen are fighting with Kariya, and one of the three brothers comes to tell Mugen they'll be waiting for him at Ikitsuki, else they will kill Fuu. Mugen is torn over whether to go and fight the brothers or stay and help Jin with Kariya. Jin then remembers his conversation with Fuu by the river the previous night, particularly her comment about Mugen (she believes that Mugen is not going to fare very well if Jin were to die and that he would wind up abandoning Fuu in one way or the other, either by his own volition or even a worse fate, by dying). He then tells Mugen to go help Fuu; he'll deal with Kariya on his own. This is a great moment in many ways: both men want to protect Fuu, and at the same time they realize they've come to respect the other after all their adventures. They also show that they trust each other: Jin trusts Mugen to save Fuu's life (which, in light of his later confession that "he used to only wield his sword for him alone, but now..." is a very big deal), while Mugen trusts Jin to stay alive and reunite with them at Ikitsuki.
Jerkass, Ax-Crazy brawler Mugen confronts the crazy brothers. Fuu insists that taking her hostage will do no good, as he is, as mentioned, an ax crazy brawler who doesn't care about, well, Fuu or anyone else. It is at this point Mugen arrives, already roughed up from fighting the first brother. In true Mugen fashion, he taunts the two remaining brothers... until one threatens to kill Fuu. Mugen offers to THROW AWAY HIS SWORD if they just let Fuu go. Fuu is SPEECHLESS. Mugen has to actually YELL at her to leave and go find the sunflower samurai aka her father.
Jin confronts and ultimately defeats, by sacrificing his own defense, Kariya (who had defeated and apparently killed him once before) to protect Fuu from his twisted version of "justice". According to Jin, it was because he finally found that something worth fighting for.
The entire scene between Jin and Mugen as they recover at Itsuki island. Mugen is perplexed about why he no longer feels like killing Jin anymore. He usually can't rest until he kills a fighter who is potentially stronger than him. Jin replies that he, too, has "found what he has been searching for." He admits that he has been alone his entire life, with Fuu and Mugen being the first friends he has ever made. Made more poignant when we realize that Mugen's only "friends" appeared to have been Mukuro and Kohza; and Fuu is also implied to be friendless as no one in her old town appeared to be significant to her and she's the only one to have never met a past connection during their journey besides the sunflower samurai. They're all each other's first true friends!