The scattering of waves in a complex medium is perturbed by polarizability changes or motion of embedded targets. These perturbations could serve as perfectly non-invasive guidestars for focusing on the targets. In this Letter, we theoretically derive a fundamental difference between these two perturbation types (the change of the scattering matrix is of rank one [two] for target polarizability changes [motion]) and identify accordingly optimal strategies to perfectly focus on the target in both cases. For target motion, at least two displacements are necessary. Furthermore, for the case of dynamic complex media additionally featuring parasitic perturbers, we establish a non-invasive scheme to achieve optimal time-averaged power delivery to a perturbation-inducing target. In all cases, no assumptions about the unitarity of the system’s scattering matrix or the size of the perturbation are necessary. We experimentally demonstrate all results in the microwave regime using a strongly sub-unitary lossy chaotic cavity as complex medium. Our experiments highlight that the target’s “structural scattering” is irrelevant [must be negligible] in the case of target polarizability changes [motion]. We expect our results to find applications in communications, cybersecurity, bioelectronics, flow-cytometry and self-propelled nano-swimmers.