  We study the superfluid response and localization dynamics from static and
mobile impurities. The superfluidity is formed in the rung-Mott phase of a
bosonic ladder model producing spin-Meissner currents induced by a
$\mathbb{U}(1)$ gauge field or a uniform magnetic field. Impurities are
described through two-state systems which act as a two-peak random potential.
An impurity sits either at the top or at the bottom of the ladder on each rung
equally, producing a telegraph signal. The impurities-matter coupling gives
rise to a classical Ising symmetry for static and mobile impurities associated
to the inversion symmetry of the two legs of the ladder. From the decoupled
rungs limit, we also identify a local $\mathbb{Z}_2$ gauge theory for mobile
impurities. The properties of the system are studied from an effective quantum
spin model including the possibility of four-body coupling in the limit of a
strong interaction between bosons and impurities. Through analytical approaches
and numerical exact diagonalization, we study the superfluid currents both in
the weakly-coupled and strongly-coupled rungs limits for the bosons. In the
weakly-coupled rungs situation, we find a smooth power-law localization whereas
the strongly-coupled rungs limit produces a steep localization or insulating
phase for various configurations of the two-peak random potential. In the
strongly disordered situation, through entanglement and bipartite fluctuation
measures, we also identify a many-body localization regime in time after a
quench of the system when prepared in a N\' eel state.
