  We unravel the correlated tunneling dynamics of an impurity trapped in a
double well and interacting repulsively with a majority species of lattice
trapped bosons. Upon quenching the tilt of the double well it is found that the
quench-induced tunneling dynamics depends crucially on the interspecies
interaction strength and the presence of entanglement inherent in the system.
In particular, for weak couplings the impurity performs a rather irregular
tunneling process in the double well. Increasing the interspecies coupling it
is possible to control the response of the impurity which undergoes a delayed
tunneling while the majority species effectively acts as a material barrier.
For very strong interspecies interaction strengths the impurity exhibits a
self-trapping behaviour. We showcase that a similar tunneling dynamics takes
place for two weakly interacting impurities and identify its underlying
transport mechanisms in terms of pair and single-particle tunneling processes.
