What role did the breakers have in the moving on of the world, and were the beams in trouble before they began their attacks, or is the whole series basically predicated on their attacking the beams and setting everything in motion?
Breakers from all over the multiverse were enslaved by the Crimson King and had their powers used to besiege the Dark Tower and attempt to destroy it, which in turn has contributed in the world moving on.
However, the process was started long before that in the time of the Great Old Ones. In an attempt to replace magic with technology, the Great Old Ones tried to destroy the source of all magic, the Dark Tower, and rebuild it. This ultimately caused great calamities and wars that ended up destroying them completely and leaving Mid-world a radioactive waste.
It sounded to me like the Breakers had been at it for a long time. Decades.
I recall that Roland mentioned that a beam went through Gilead. By the time of the end of the story, only two beams out of 6 remain. I think it likely that a beamquake may very well have coincided or perhaps even finished off Gilead entirely after the conflict with the Good Man.
Since before the fall of Gilead they had been saying that the world was moving on, I would surmise that the Breakers had been working (in one capacity or another) since then. Roland describes in Wizard and Glass that the revolution with the Good Man was a symptom, the disease was elsewhere causing the fall of civilizations and affecting time and space. He had heard of thinnys before coming to Mejis, but had never seen one, but they were present and common enough (although still rare) that they were not unheard of.
So, I think the Breakers had been causing problems for a very long time. Ted only came to them recently (I am thinking that he was only there for a few years, perhaps 10? With Ted on board, they quickened their pace of beam destruction by an order of magnitude or more.
In the grand scheme of things, I would say that the Breakers are the last act, the final thrust in a long game that has since finally come to blows. Throughout the books they mention the game of castles which sounds similar to chess played with some form of obstable (a hillock) to shield players from one another. It is often played such that players move their peices into position behind their hillock, and circle, not exposing themselves until they are ready to finaly take their enemy apart.
So to, do I think that this fight to end the world has been conducted. North Central Positronics, Lamerk Industries/Foundry, Sombra Corporation where the long ago efforts to start the fall. They were the prep stage. Once the Imperium (and whatever forces they were fighting) fell and Lud and Dis fell into ruin, other agents (Flagg being chief among them) went about sowing discord and felling kingdoms and baronies until only a few remained. Arthur Eld and the Affiliation where an effort by the White to re-establish order and light, but that to eventually was countered.
Toward the end, with most of the other players out of the way, an assault on the tower itself was possible, and the Breakers were brought in to play to perform it.
The Red has been working on this for a long time, and the Breakers are the last hammer blow executed to finish it. While the Breakers may not have been responsible for the problems in the world in the long ago, their activity certainly is contributing mightily in the problems of the "current day" in the story.