Think of a Falcon 9 second stage but instead of payload, it had more tank, more fuel, a docking ring and a set of RCS thrusters. It gets to orbit, finds its ”passenger”, docks to it and pushes it into whatever trajectory it wants to go.

Add a truss with multiple docking rings to your payload and you can add multiple tugs to give a larger payload a bigger kick.

Even packing a few kick stages on top of F9s or F9 heavies would enable some interesting missions - as both trajectory can be higher energy and payload doesn’t need to be trimmed to the minimum viable mass.