Almost all old furniture is finished with either shellac or lacquer (shellac until the 1920s, lacquer since). Both of these finishes dissolve into a messy “gunk” when paint stripper is applied. On complex surfaces, such as turnings and carvings, an easy way to remove the gunk is to wipe it off with handfuls of wood shavings from a jointer or planer.
The shavings soak up the gunk and you can remove them easily with a stiff bristle brush.
If you don’t have a planer or jointer of your own, you may be able to find the shavings at a local woodworking or cabinet shop.