As long as the temperature inside remains constant, as much cold leaks out as is transported inside. So the only residual heating outside would be from inefficiensies in the system, not the moving process itself.

To be extra clear: An AC transports the heat, not the hot air. It removes heat from the air and transfers that heat to the outside air.

There's also heat pumps that work with water instead of air. So they remove heat from the air and push it into water. This water can be a closed loop, or be open where the water is lost. It can also work the other way around where the heat pump takes heat from outside and pumps it into water, heating up the water to then be used for heating a home or taking a shower. There are also water-water pumps that work on water on both ends.

Because heat pumps pump the actual thermal energy, the medium doesn't really matter much.

You need some lessons in thermal dynamics my man.