  For two self-adjoint operators $H,A$ we show that a general commutation
relation of type $[H,\mathrm{i}A]=Q(H)+K$, in addition to regularity of $H$ and
Kato-smoothness of $K$, guarantee pointwise in time decay rates of diverse
order. The methodology is based on the construction of a modified conjugate
operator $\tilde{A}$ that reduces the problem to previously developed estimates
when $K=0$. Our results apply to energy thresholds and do not rely on resolvent
estimates. We discuss applications for the Schr\"odinger equation (SE) with
potential of critical decay, and for the free SE on an asymptotically flat
manifold.
