Play a Rite
A Rite resolves the moment you play it and then leaves. Drift moves a body to the next lane and Readies it — which is how the Cogum you just summoned gets to act on the turn it arrived.
The board plays at its own pace — the rival takes its turn where you can watch it. Advance the phase and end the turn from the rail on the board’s edge, exactly as you would in a real game.