If you are not satisfied with the paint job you've done on an older miniature, all is not lost. Pick up a large bottle of Pine-Sol brand cleaner, the golden-colored variety works great. I would recommend against using an off-brand or generic cleaner, which may not strip paint as effectively.
I have not used the more pleasant-smelling blue variety of Pine-Sol. My wife bought some the other day, hoping that it will work without stinking up the house. I'll let you know what I find out. The normal stuff is a bit strong-smelling, so open a window or otherwise ventilate the area where you will clean the paint off your minis after soaking them in Pine-Sol.
Pine-Sol, as the name suggests, uses pine oil as the active ingredient. This stuff attacks paint aggressively. In fact, if you strip miniatures attached to a plastic base, such as Games Workshop's "slottabases," remove them before cleaning with Pine-Sol. The bases can actually go soft on you.
Get a glass jar with a lid that seals. An old pickle jar works well, if cleaned out well. Fill the bottle most of the way full of Pine-Sol. Carefully place miniatures inside, completely submersing them. Screw on the lid tightly to keep the odor to a minimum. Let them soak for a full day.
After soaking for about 24 hours, you'll find that most paint ready to remove. It doesn't matter if you need to remove acrylic, oil paint, enamel, or primer. Pine-Sol strips them all. It attacks the paint, making it soft so you can scrub it off.
You'll want to work in a well-ventilated area. If you can work outdoors, this will keep the odor down indoors.
If you have sensitive hands, you may want to wear latex gloves when scrubbing off the old paint. Pine-Sol can dry out your hands and you'll smell like pine for hours afterwards if you don't.
Use a hard-bristled toothbrush to scrub away the soft paint. Use a small drop of liquid dishwashing soap on the toothbrush. This is necessary, since you not only need to get the paint off, but you must remove the pine oil in order for your primer to stick to the miniature prior to applying colored paint.
If you can't remove all paint from your miniature the first time around, soak it for another day and use the same process to remove what's left. Especially old enamel paint may take a second scrubbing to remove all paint, especially from deep crevaces and indentations in the miniature.
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I have successfully removed paint from plastic miniatures in the past without causing damage to them. That said, I would take care not to leave plastic or resin miniatures soaking in Pine-Sol for more than an hour or so without checking to make sure that it doesn't damage your miniatures.
I have seen some forums which advocate using brake cleaner and other aggressive chemicals. I have not tried these, but I've heard they can be pretty nasty to get on your hands. Besides, Pine-Sol works great, and fools people into thinking you cleaned your house. :)