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”
• “Women tend not to advance in the major law firms”
• “I got promoted after many years of hard work”

make publicity for; try to sell (a product)

• “The salesman is aggressively pushing the new computer model”
• “The company is heavily advertizing their new laptops”

be changed for a superior chess or checker piece

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