I'm not exactly sure how the forces worked to bow the rim carriers out or what actions started first then next then next. And for the correction, it may not matter much, but I'll speculate based on my observations in a minute.
But as to the correction, once the caved joists are fully jacked back up, the contractor is going to bolt 2x16 glue lam to the barn posts (12' o.c.) on a plane that will be under the joists, so as to no longer rely on the joists hangers or the existing carriers alone. In addition, he will run supports up to the carriers from the stall posts that form the opening for the stall doors to help stiffen and support everything.
Now, here's my speculation about the existing carriers and failure. Mind you the carriers were made up of one 2x8 and one 2x10, nailed to the inside of the posts spanning 12'. I believe they first bowed down, possibly enough to lengthen the middle joists space beyond the length of the joists or at least without enough bearing surface to hold. Once one joists drops, it would act against the hanger, pushing it and the carrier outward, then causing, the 2 neighboring joists to slip off the hangers because of the ever expanding span. This would continue outward from the center until it progressed closer to the posts which did not allow further expansion of the span.
Then the outward and downward motion stopped and was supported be the plywood nailed to the joists and somewhat supported by a run of large (1.25"?) Conduit that had a 90 degree bends down to boxes which formed an arch of sorts.
There may have been a different sequencing, but that is my guess based on my observations.