promote

contribute to the progress or growth of

I am promoting the use of computers in the classroom

give a promotion to or assign to a higher position

John was kicked upstairs when a replacement was hired

make publicity for; try to sell (a product)

The salesman is aggressively pushing the new computer model

be changed for a superior chess or checker piece

change a pawn for a better piece by advancing it to the eighth row, or change a checker piece for a more valuable piece by moving it to the row closest to your opponent