Rules

A ball is putted and rolls off the green and hits a push cart:

Under Rule 11.1, for accidental deflections, including when the ball hits the player or opponent or their equipment or caddies: There is no penalty and the ball is played as it lies (with limited exceptions).

If a ball is accidentally moved while searching for it

If you move any player's ball (even your own) while searching for it, there is no penalty, just replace the ball.