I can speak to the conflict between Congregationalists and Unitarians, at least. I'm aware of a substantial amount of tension between Methodist missionaries like Peter Cartwright and Universalist groups on the early 19th century "frontier" (Ohio at the time), but doesn't quite get to your question. The Unitarians and Universalists didn't merge until 1961, but perhaps there were some other Congregational-Universalist conflicts that I'm not aware of.

If you go to Cambridge, Massachusetts you'll see a pattern that's repeated in many other New England towns. First Parish in Cambridge is just across the street from Harvard Yard, and First Church in Cambridge is a few blocks away from the town center. Obviously the question is which is the real first church.

The answer goes back to a court case in Dedham, Massachusetts in 1818. But some background first: according to its members, the first Unitarian church in the United States was King's Chapel in Boston. That was a bit of an outlier in the Unitarian movement, however, as the Unitarian leanings of the church were more closely connected to one pastor's personal revelation than a broader movement. The real Unitarian moment in New England didn't come until the early 19th century. Charismatic Unitarian preachers claimed that an academic reading of scripture led to the inevitable conclusion that God was a unity rather than a trinity (in addition to a collection of other conclusions), and the message spread rapidly. You can take a look at William Ellery Channing's Baltimore Sermon if you're interested in what sort of message these ministers were spreading at the time.

Unitarian converts started to take over Congregationalist institutions. Traditionalists panicked as a Unitarian was appointed to Harvard's Hollis professorship in 1805 -- which led directly to the founding of Harvard Divinity School. HDS became Harvard's first independent graduate school specifically because university leaders wanted to reassure traditional Congregationalists that their sons could still receive an undergraduate education untainted by Unitarian ideas -- which would be confined to the Divinity School slightly off campus. Still, many parents opted to send their children to Yale instead, which retained its status as a bastion of old-style New England Congregationalism.

That institutional takeover went down to the church level as well, which brings us back to the churches by Harvard Yard. Congregational churches were largely independent from one another without any larger governing hierarchy. Ministers were chosen by parishes (everyone who lived in town). Therefore, when a sufficient number of a townspeople turned Unitarian they then had the power to elect a Unitarian minister -- regardless of the wishes of the church's actual members. Obviously, that led to some conflict.

In 1818, when the people of Dedham elected a Unitarian minister for the town's church, its Orthodox members protested by taking up all of the church's documents and valuables elsewhere. They insisted that they alone had the right to decide the leadership of their church, despite the fact that the church was supported with public funds. There were further arguments and trials, which there's a bit more on here. But the upshot is that the Unitarians won and the traditionalists had to leave and start their own church.

So First Parish in Cambridge is Unitarian -- the original church taken over by the Unitarian townspeople in the early 19th century. First Church is Congregationalist -- the traditionalists who split away and moved down the road. Of course, there were impassioned theological debates taking place alongside all this, but that's a subject for another post.