  The topological Kondo effect is a genuine manifestation of the nonlocality of
Majorana modes. We investigate its out-of-equilibrium signatures in a model
with a Cooper-pair box hosting four of these topological modes, each connected
to a metallic lead. Through an advanced matrix-product-state approach tailored
to study the dynamics of superconductors, we simulate the relaxation of the
Majorana magnetization, which allows us to determine the related Kondo
temperature, and we analyze the onset of electric transport after a quantum
quench of a lead voltage. Our results apply to Majorana Cooper-pair boxes
fabricated in double nanowire devices and provide nonperturbative evidence of
the crossover from weak-coupling states to the strongly correlated topological
Kondo regime. The latter dominates at the superconductor charge degeneracy
points and displays the expected universal fractional zero-bias conductance.
