This is based on opinion. Please don't list it on a work's trope example list.
Tearjerker / Liberty's Kids
Cato and Moses's backstory. As children, during a typical part of the slave trade, they were separated. Moses worked until he earned enough money to buy his freedom, but Cato was passed from owner to owner. Their ongoing arc is one of many that has a sobering conclusion
When Moses tries to buy Cato at an auction, with Sarah generously offering to pay, the auctioneers refuse because Moses is a black men. Moses then tries to break Cato out, only to get caught and is only saved when Sarah, James and Henri vouch for him.
Cato manages to run away and join the British side. When he keeps his mouth shut after realizing that James Armistead is a spy for the Continental Army, he reacts with surprise on hearing that James knows about Moses.
The British then leave the slave recruits behind when they realize they're losing Yorktown. Disgusted by this, a Hessian runs into the battlefield and orders all of the survivors, including Cato, to Come with Me If You Want to Live. He then has to Shoo the Dog after the battle and wishes Cato luck.
Cato is then missing for years with only a wanted poster as a clue. He then comes to Moses as a fugitive; while they are happy to be together, Moses knows that their reunion is short-lived. He decides to send his brother with a Loyalist woman to Canada, where he can be free.
Sarah's attitude towards slavery. When she finds out that Phyllis Wheatley is a slave and a poet, she talks about how the slave trade is dying in Britain but thriving in the colonies. For the rest of the series, despite her Conflicting Loyalty about being a patriot or a loyal British subject, she remains a staunch abolitionist.
The death of Nathan Hale. Even James cries in that scene.
James also nearly becomes impressed onto a prison ship. Just because he happened to be in the wrong place at the wrong time.
United We Stand: A mob tars and feathers a sailor named Mr. Parker, and James, thinking that the man got what he deserved for being unpatriotic, joins the crowd to laugh at the poor man's misfortune. When he expresses a desire to add an article on the event to the paper, Moses decides that James has to see what became of Mr. Parker. At the doctor's, James is treated to a rather graphic description of what happens when you're tarred and feathered followed by an equally explicit description of the side effects of the tar removal (namely the peeling away of skin and the subsequent infection of the resulting wounds), taken into the room to see the sailor covered in bandages and tearing up in pain, and then told that the tears cause an unending cycle of pain because the salt in the tears only makes that man hurt more when they leak into the wounds. James starts crying.
Moses: I brought the young man I mentioned earlier. Can he ask you a question or two?
What makes this even worse is it's unclear if he survived or not.
Two words: Cousin Tom. Sarah's tears during that scene say it all. Not to mention her narration afterwards:
Sarah: Oh, mother, Tom's death seems to me a sign of awful things to come. This conflict is not at all what James thought it would be. It won't be simply contest of ideas. It promises to be a contest of arms. It promises to be everything I had prayed we would avoid.
The death of Dr. Joseph Warren at Bunker Hill; James had worked frantically to see if he had made it out alive, only to find out he died covering the American retreat.
In the same episode, we see the results of the battle on the British side; nearly half of their men had been killed or wounded, and many of the junior officers are resentful that so many of their comrades had to die for such modest gains.
Among the dead, a certain lieutenant who personally knew Sarah's father and where he might've been; Sarah instead finds the lieutenant's brother who informs her of his death. Sarah is left to console the heartbroken soldier upon hearing the news.
The chaos on Bunker Hill is so much that Henri has lost his appetite by the time the gang has moved away from the frontlines.
Colonel Rall's death at the Battle of Trenton, where he was caught completely unprepared by Washington's attack and was wounded in the engagement. Worse still, he had been given a written warning from a local loyalist that he had seen Washington cross the Delaware, but neglected to read it. After seeing what the note entailed, he can only lament how his own carelessness had rendered himself helpless and approaching death. Truly a case of Alas, Poor Villain. While we do not see his fate onscreen, the real Colonel Rall would succumb to his injuries shortly after personally surrendering to General Washington.
The real-life fate of Cornstalk and his son in the episode The New Frontier. They're detained at Ford Randolph, even though Cornstalk's tribe was neutral, and then they're wrongfully executed. Sarah is so horrified by what happens to them that she outright leaves the colonies and returns to England.
The reactions of the other main character's to Sarah's departure in the same episode can be seen as a Tear Jerker. James is so upset that he refuses to say goodbye to her, and is still somewhat bitter when she returns three episodes later. Henri, like some viewers, perhaps, is in silent tears during the scene. Walter Cronkite's solemn narration as Dr. Franklin really helps, as well.
The attack on Fort Tryon, where Sarah befriends the real-life Molly Corbin. The two witness the death of Molly's husband by cannon fire, and Molly herself is injured trying to reset it. The only thing Sarah can utter as the British overtake the redoubt is "God, help us".
It is an especially big relief to James to see that Sarah and Molly both survived their ordeal, after what proved a disastrous battle for the Americans.
Speaking of which, the fall of Fort Washington, where General Washington himself takes full blame for his failure to abandon the fort in the face of a superior force, leading to the loss of thousands of men.
The very early scene where the group has met Phyllis Wheatley and Henri reveals his backstory as a slave: his parents died on the trip to America and the captain forced Henri to pay off their debt. It's obviously a painful memory given the bitterness in Henri's voice and Sarah's disgust is quite apparent (albeit subdued) in her declaration of "How dare he...". Henri's response to that ("I belonged to him.") doesn't help matters and it positively hurts to see the flashback of Henri being terrified of James and Moses because he thought they wanted to hurt him.
James' backstory is that he was orphaned as an infant when his parents died in a fire caused by a lightning strike. He sought out Benjamin Franklin when he was old enough, but quite likely lived on the streets or at least in an orphanage until then.
The betrayal of Benedict Arnold after his attempt to sell out West Point to the British; Although more focus is placed on the outrage surrounding the affair, it is especially saddening if you look at it from the perspectives of Washington and Sarah, the former who had defended the general at every turn and the latter who was motivated in part by Arnold's bravery and gallantry to join the patriots' cause. All through their time together Arnold had treated Sarah especially graciously, and their last meeting saw him fleeing the scene, not even bothering to refer to her by name. Sarah's admiration of him is immediately replaced by the same revulsion expressed by everyone else after his treachery.
In "The Man Who Would Not Be King", Washington is at odds with many of his former comrades at the war's conclusion, including General Horatio Gates who had long held bitter jealousy of Washington. Fearing he would install himself as a king now that their victory was assured, they gang up on him at a tavern. Washington however calmly addresses the men, quietly dismissing their fears while repeating his conviction to the ideals of liberty he devoted his service to. Before reading a prepared statement, he pulls out his bifocals and puts them on, much to the shock of his officers who had never seen him wear them in public. Moved by this simple gesture, they lower their defenses and let the general voice his thanks to them.
"In Praise of Ben"
Sarah is worried that Henri, James and Moses will think she's "a silly girl always changing her mind" since she left them and has decided to return. It becomes a Crowning Moment of Heartwarming when Moses pulls her into a hug and James returns her hug reluctantly.
Henri gets into a fight with another boy named Charles for insulting "Dr. Franklin". Moses pulls them apart and brings them inside to reach a more peaceful resolution.
Charles has a lot of mistaken beliefs about Ben, including that he had a "fancy education". Moses tells him otherwise: Ben had to leave school when he was seven because his family couldn't afford the fees. Apprenticed to make candles, Ben would use his earnings to buy books and study late at night.
Charles's father then comes in and is a Jerkass. He tells Moses that he doesn't like seeing slaves free.
Henri and Charles are nearly trampled by a runaway horse, if not for Moses's quick thinking. Charles's father nuzzles him while Sarah hugs Henri.
Henri: Sarah, I almost didn't hear about your journey back.
Sarah encounters a young soldier named Robert Shurtlieff who seems too young to shave. Then Robert gets shot during a skirmish, and insists on treating his own wounds. Sarah is intrigued and worried. Robert then lapses into unconsciousness from illness, and the doctor opens his shirt, revealing bandages. He and Robert both go Oh, Crap! as Robert hurriedly whispers to him.
Sarah's return home to England, during which her vessel is caught in a violent storm and sunk; Sarah is the only survivor of the ordeal.
After John Paul Jones's crew rescues her, he's delighted to recognize her from her articles. Sarah confesses that she hasn't written in a while, since Cornstalk and his son's unfair execution, and takes a while to open up.
John Paul Jones mourning the loss of the Bonhomme Richard after capturing the Serapis. You can hear his voice break as his beloved vessel sinks beneath the waves.
John Paul Jones: She was a cast-off ship with a cast-off crew, Ms. Phillips. Yet she beat the best. And....she was mine.
James meets the Mohawk leader Joseph Brant, who sided with the British to protect his people from further colonial spread. He asks James point-blank what his people are supposed to do with either choice in allegiance guaranteeing their deaths.
When Congress is debating many issues, slavery is an immediate hot-button. No one can agree on what to do, with some hateful language being thrown around especially the slave owners advocating for abolition. Moses is especially worried as a freed slave whose brother went into hiding.
Ben Franklin breaks the news to Moses that Congress won't abolish slavery. Moses is heartbroken, and asks And Then What?. He's a peace-loving man, but he doesn't know how the conflict can resolve. Ben Franklin agrees; the country had just fought one war but it may take another for abolition to hold.
Moses decides that he will form a school for black children to learn to read and write. Ben Franklin offers all his support. Given how such institutions are the targets of hate crimes, and would later be temporarily limited by laws . . .