... if you're certain the colors won't run, I'd wash it like washing old yarn. Make a soapy bath with a baby shampoo, dip the piece in and out. Use a makeup brush (we need soft bristles) to gently work at any bad spots. Make a clean water bath and be prepared to change it a couple times, dip the piece repeatedly and change the water frequently until it's soap free.

I've heard, but never tested, that you could try and fix the colors with a salty water spritz before you wash. I don't recommend it because I haven't tried it, but it had been recommended to me. You can also try to sandwich the piece between some screening or other fabric to minimize the chance of threads coming loose, but I don't think that's likely if you're as gentle as possible with the lifting/brushing. Nothing a little stitch fix wouldn't mend in the worst case.

When you're done, there's archival glass displays you could put that in to prevent future dust and grime as well as light decay.