How do I keep painted faces from smearing when I add clear glaze over them?

The light re-fire is a real technique and it's probably your best fix here. After you paint the details on the bisque, fire the piece again at a low temprature, cooler than your bisque fire. This hardens the underglaze so it stops behaving like wet chalk when you brush over it. alot of people who do detailed brushwork do this as a matter of course. It uses up a little kiln time, but if crisp outlines are the whole personality of the piece, it pays for itself.

If you don't want to squeeze in an extra firing, your thin-wash-of-clear-first idea also works, just with a caveat. The very first coat of clear is the one that does the smearing, because it's re-wetting the underglaze. So apply that first coat as thinned-out as you can get away with, and use a big soft brush so you can lay it on in one or two passes without going back over anything. Every pass over a half-dry line is another chance to drag it. Let that coat dry completley, then apply your second and third coats normally and the lines will be locked in.

A couple other things that helped me with the same problem on sgraffito-style line work. First, thin your clear glaze slightly with water, maybe the consistency of light cream instead of yogurt. Thicker glaze grabs and drags more. Second, one-directional strokes only, top to bottom, and resist the urge to smooth or touch up. It feels wrong, but the glaze self-levels in the kiln and your brush marks disappear while the smears don't. The spots where you've layered colors, like the cheeks and ears, will always be the most fragile because layered underglaze stays softer longer. Hit those areas with a hair dryer for a minute right before glazing. Dry underglaze is much more stubborn about staying put. And honestly, don't accept the melt. Your cat deserves his edges back.

Agreeing with the hair dryer tip on the layered spots, that's the part people skip. I do the same thing on my little bird bowls where the wings overlap the body color, and I also hold them up to a lamp after drying. If the underglaze has any shine left to it at all, it's still damp enough to drag, so I give it another minute with the dryer before glazing. Made a real difference on the fine feather lines. Your cat will hold his face, trust me.

Lurker here, first post actually. I paint designs on stones and seal them with varnish, and the first coat dragging the paint around is the exact same headache, so this thread is weirdly familiar even though I don't fire anything. What saved me was sacrificing a couple of test pieces before touching the good ones. Do you have any warped or ugly bisque pieces lying around you could try the second coat method on first? Also curious if anyone knows wheather the age of the underglaze matters, mine sometimes seems thirstier when the jar's been open a while.