It really depends on the manufacturer of the stroller, but a lot of the fabric parts can be taken off and washed in the washing machine and then thrown into the dryer.
My daughter received a hand-me-down stroller that was mildewed and filthy and we were able to take all the fabric parts off, wash it in a bleach hot water wash, then put them into the dryer. After she cleaned and disinfected the frame we placed the fabric parts back on.
I just bought a beautiful stroller at a yard sale that looks like new, but saw a bit of dirt on the fabric once I got it home. I did the same to this stroller as we did to the other stroller. I had to unscrew a few things to get the fabric pieces off, but it is so clean now and looks so beautiful!
Try taking the fabric pieces off and washing them in hot water with bleach in the washing machine. If you are afraid of shrinkage, then let them air dry for a couple days.
at 4:59 PM on Jun. 6, 2009