  We examine the response of a system localized by disorder to a time dependent
local perturbation which varies smoothly with a characteristic timescale
$\tau$. We find that such a perturbation induces a non-local response,
involving a rearrangement of conserved quantities over a length scale $\sim \ln
\tau$. This effect lies beyond linear response, is absent in undisordered
insulators and highlights the remarkable subtlety of localized phases. The
effect is common to both single particle and many body localized phases. Our
results have implications for numerous fields, including topological quantum
computation in quantum Hall systems, quantum control in disordered
environments, and time dependent localized systems. For example, they indicate
that attempts to braid quasiparticles in quantum Hall systems or Majorana
nanowires will surely fail if the manipulations are performed asymptotically
slowly, and thus using such platforms for topological quantum computation will
require considerable engineering. They also establish that disorder localized
insulators suffer from a statistical orthogonality catastrophe.
