Since we paint every week, we usually have at least one wet painting to deal with. My kids like to cover paper with paint, so the painting is messy and droopy when I try to move it. A while back we started transferring paintings from the easel to a piece of newspaper. (If painting at the table, the paper is placed on newspaper before the kids begin to paint.)
The newspaper gives us clean edges to hold when carrying a painting, no matter how much paint was put on the paper. The newspaper also gives us a place to print a name and makes identifying paintings easier when going home.
We also use the newspaper as a "portfolio" when sending the painting home; this comes in handy when a painting is still wet. We turn the painting and fold the newspaper closed, with all the wet paint inside the newspaper. Parents or kids can carry the painting around without trying to balance it or keep it from touching clothes or belongings. Sometimes paint is transferred to the newspaper, but the painting remains relatively intact and travels home safely.
(If you're going to paint a lot, make the trip home easy for parents!)