The Overwatch game mode in which an attacking team escorts a payload is called Escort.
In an Escort match, one team attacks by moving a payload along a fixed route toward a delivery point, while the defending team attempts to stop its progress. The payload advances when attackers remain near it, stops when defenders contest it, and can move backward if attackers are absent and no checkpoint has secured its position.
Passing checkpoints usually adds time to the round and prevents the payload from retreating beyond that point. This creates shifting battles: attackers must clear defenders and stay with the vehicle, while defenders look for opportunities to eliminate attackers or force them away from the route.
The similarly named Hybrid mode is a frequent source of confusion. Hybrid maps begin with an Assault-style objective: attackers must first capture a location containing the payload. Only after that capture does the Escort phase begin. Control is different because teams fight over a stationary objective, while Assault focuses on capturing fixed points. Thus, when the question asks for the dedicated payload-escort mode, Escort is the canonical answer.