I just completed a Lego table for my daughter. I sprayed flat oil based paint on it. My original intention was to put the paint on and then wipe it off giving it a stained look then lacquer over that and I have done that with success in the past. The paint wasn't wiping off well so I decided to just paint it. I could not find satin in the store because they only carried flat, semi-gloss and gloss. I bought the flat thinking I would just spray some medium rubbed lacquer over the paint. I think this should be fine, but just wanted to run it by the experts just in case.
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Wish I had researched... The guy at Sherwin-Williams said that it would cause the paint to wrinkle. Said I should go back over it with a semi-gloss oil based paint. I guess I'll do that to be safe. I just wanted something more durable since it will be in the playroom.
Lacquer would have the propensity to distort oil base paint. You could do a light sanding and use an oil base varnish, or oil base polyurethane. Whatever you decide, allow each application to completely cure before re-coating.