I just spent about half an hour going around my house and polishing up all of my encaustic paintings (I have a lot on my walls). Because encaustic paint is wax, it does this thing called “bloom”, where some of the moisture in the wax slowly escapes and settles on the surface. It can make the surface of the painting look dull.
It's a subtle difference that a lot of people wouldn't even notice, but I really prefer a polished painting than an unpolished one.
Bloom is really only a problem on paintings with little to no surface texture. You generally don’t notice it on heavily textured paintings, and you never want to rub a textured painting because they’re pretty fragile.
If you have a pair of Hunter rubber boots, you’ve probably seen bloom before – especially on black ones. I clean my wellies every couple years to get rid of the cloudiness that happens because of bloom. I'd do it more often, but I wear them in the mud so it's not like I'm super concerned with how clean they are...
Anyway, back to the paintings. First, use a feather duster or something else gentle to brush any dust off the surface. I also sometimes just blow on a painting to remove dust.
Then, all I do is get a super soft, clean, dry rag (I use a cut-up old t-shirt or flannel) and gently rub back and forth across the face of the painting with the rag over my fingertips. You have to be careful not to gouge the surface or press too hard. But it’s pretty simple – just rub till it’s nice and shiny. The polishing will make the colors vibrant again and get rid of any fingerprints that have been left on the surface.