Does the dishwasher rinse the dishes with post-filter grey water?
Yes. The dishwasher fills the bottom basin with water (and heats the water to a good bacteria-killing temperature) then pumps the basin water to the sprayers. The dirty water falls back down into the basin where it is reheated and pumped up again. The dishwashing detergent helps get the dirt off the dishes and keep it in the water. The dishwasher empties and refills the basin several times during the whole cycle, so the spray water gets progressively less dirty. Eventually, by the final rinse, the water is clean enough to leave the dishes clean. (The fact that the dirty water is used to wash the dishes can be why the dishwasher instructions say to pre-rinse the dishes of major food pieces before putting them in the dishwasher. Then the spray water won't be as dirty as un-pre-rinsed dishes would make it.)
how did the food particles get through the filter?
There's a hole. Somewhere. Or the unbroken filter mesh hole size is enough to stop green beans, for example, but not corn kernels.
My old dishwasher had an open basin, a grid-work cover over the pump sump, and a filter in the sump whose mesh hole size was large enough to allow corn kernels through. The pump shaft extended through the filter and had a 'shredder' (a piece of stiff wire) on the end which would smash up larger food pieces that got through the grid cover. We'd get corn kernels and peas in the spray arms all the time. (I guess we weren't pre-rinsing the dishes enough.)
My newer dishwasher has a large filter plate over the whole basin with much smaller holes. We also are pre-rinsing the dishes better. Haven't had clogged spray arms in a long time.
