Method One: Using Spit
Buy several cotton swabs. Using your spit to clean a work of art might seem both crazy and profane, but it's a method museum curators around the world use. Saliva has enough enzymes to break down dirt and grime, but not so much that it will damage the paint.[1] It'll take a lot of cotton swabs, though, so stock up with a full box.
Be aware that this method takes some patience — ask a museum intern
how many hours it can take to clean a large painting this way. (Hint:
It's more than 2.) But if you want to meticulously clean your painting
with almost zero risk of damage to the oils, this is the way to go.