Here's how I apply shake & dry polyurethane. There may be other ways to do it but this method works for me.
1) For gas check bullets, I recommend installing the check and sizing the bullet prior to coating, otherwise the coating will increase the shank diameter and then the check may not want to go on. Wash your sizing die & push stem with solvent first to remove all traces of lube so the bullet does not get lube on it. If you are planning to heat treat the bullets then heat treat prior to coating (so far I've had best luck with air-cooled WW for coated plain base bullets and heat treated WW for coated GC bullets, your mileage may vary!)
2) add thinner to the polyurethane ahead of time. You won't use much polyurethane so only mix a few ounces at a time. A solvent-compatible squirt bottle is very convenient for holding the mix, like mustard bottles or hydrogen peroxide bottles.
3) for oil-based polyurethane, thin with 30% - 50% paint thinner. Paint thinner dries slowly, but unfortunately acetone does not seem to be compatible with oil-based polyurethane. Perhaps there is a compatible thinner that is faster drying but I haven't found it yet. You don't absolutely have to thin, but un-thinned poly will result in a thicker and more uneven coating.
4) For water-based polyurethane, thin with either 30% acetone (fast drying) or 30% water (slow drying).
5) Put the bullets in a solvent-compatible tub. Either soft butter tubs or cool whip tubs work well. You need to be able to swirl the bullets around so small batches work best. Add 1/2 - 3/4 teaspoon of poly mix and swirl until the bullets are evenly coated. You want just enough poly to coat the bullets but there should not be excess liquid on the bottom of the tub. If necessary, dribble a little more poly in until all the bullets are 100% wetted. If you add too much poly it's not the end of the world but it will take longer to dry.
6) Swirl the bullets in the tub for about 2 minutes. With acetone-thinned water-based poly, the coating will be tacky after only 2 minutes. With paint thinner or water thinner the coating will still be wet, but that's OK, go ahead and dump the bullets out onto a wire tray.
7) The wire tray was made from 1/4" hardware cloth. Place the wire tray in a warm spot. I put it behind my wood stove, perched on a couple of pieces of kindling so air can circulate.
8) Every few minutes, gently shake the wire tray. This prevents the bullets from sticking to the tray or to each other as they dry. For acetone mix, you usually only need to shake the tray one time. For slow drying mixes I usually end up shaking the tray 3 times. After that the bullets are dry to the touch and can be left alone.
9) Let the coating cure 30 - 180 minutes, then apply a 2nd coat.
10) I prefer 3 thin coats. They usually end up a total of 0.0006" - 0.0010" thick. If it's more than 0.001" thick then your mix is too thick and your coating is likely to be uneven and unbalanced. If it's less than 0.0006" thick your mix is too thin. That doesn't hurt anything other than you may need additional coats to compensate.
11) It is possible to speed up the cure time by heating in an oven at 200 - 250 F for a few minutes. But don't do this on heat treated bullets because it reduces the hardness.