first check the batting package and see how it says it can be laundered- i've used dream wool which is totally wash/dryable.
but different batts have different requirements.
i tend to wash/dry my quilts exactly like i think they will be cared for by the person i'm giving them to-for baby/kids quilts i expect them to be tossed in to the washer and the dryer-so that's what i do when i finish one- i toss it in --usually with warm water because thats what i tend to use- a regular cycle- then i toss it into the dryer- if it doesn't survive that it's not going to survive a child/baby using it---
i actually wash/dry all of my quilts this way-unless it's a wallhanging- item i do not expect to ever be washed....in the 35 years of making quilts i've never made one that did not go through the washer/dryer -- some i don't wash/dry often---like the wool quilts that normally just need to be vacuumed/ spot cleaned/ or taken out & dusted/ shook out-
the silk/velvet/crazy quilts were washed when finished- but i did lay them outside on a sheet(draped over my hammock) with another sheet over it to protect it from birds-to dry