Stroboscopic measurements of a time-dependent projector generates a quantum Zeno dynamics whose effective Hamiltonian is the Kato-Avron Hamiltonian for parallel transport, which reduces to counterdiabatic driving in the case of a nonselective measurement of the instantaneous eigenprojectors of the system. The same generator arises under continuous monitoring of a time-dependent observable and under a time-varying complex absorbing potential. Departures from the ideal Zeno limit are governed, in all three settings, by a single positive leakage operator with a dynamical and a purely geometric contribution. Matching the corresponding generators yields a model-independent generalized Schulman relation for the quantum Zeno dynamics in the three scenarios, and a purely geometric bound on the measurement resources needed to steer a subspace along a prescribed path.