Any stain will come off a bit when any solvent, be it water or alcohol or oil is applied. The solvent is only there to get the dye pigments into the wood. Once the solvent has evaporated, the pigments are fairly free to move around if another solvent comes along. It's just that alcohol/spirit stains tend to penetrate the wood a bit more, so are less likely you spread as much. The trick is to get the first application (or two) on quickly and don't worry too much about smoothing it out, as long as it covers the surface. Once you get a good layer or two on, the stain is sealed in.

Some people apply the first coat of finish with a brush. If you have a stained burst, then always brush outwards from the lighter to the darker shades.

With a single colour stain, even wiping on Tru-Oil shouldn't be a problem (I've only done it once but it worked OK for me).

Otherwise, a spray can for your first coats are the best solution, so there's no physical movement of any stain absorbed by the finish.

If they aren't stains designed for use for wood, then that may make things worse.