  The motion of a tagged degree of freedom can give important insight in the
interactions present in a complex environment. We investigate the dynamics of a
tagged particle in two non-equilibrium systems that consist of interacting
run-and-tumble particles. The first one is an exactly solvable polymer model,
the second is a two-dimensional lattice model, which is studied through
simulations. We find that in both cases a tagged particle shows anomalous
dynamics and non-Gaussian behaviour for times below the persistence time of the
run-and-tumble motion. For later times, the dynamics of the tagged monomer
becomes diffusive and Gaussian. In the lattice model, non-Gaussianity persists
and can, for intermediate densities, be well approximated by a Laplace
distribution. We attribute this behaviour to the dynamically changing
environment of the tagged particle, which we argue, is an essential ingredient
to observe deviations from Gaussianity.
