  We review a time-dependent version of the Schrieffer-Wolff transformation
that accounts for real-time control of system parameters, soon to be rendered
possible on a broad basis due to technical progress. The dispersive regime of
$N$ multilevel systems coupled to a cavity via a Jaynes-Cummings interaction is
extended to the most general case. As a concrete example we rigorously apply
the technique to dispersive two-qubit gates in a superconducting architecture,
showing that fidelities based on previous models are off by up to $10^{-2}$,
which is certainly relevant for high-fidelity gates compatible with
fault-tolerant quantum information devices. A closed analytic form for the
error depending on the target evolution closes our work.
