  We study a quantum quench in which a magnetic impurity is suddenly coupled to
Hubbard chains, whose low-energy physics is described by Tomonaga-Luttinger
liquid theory. Using the time-dependent density-matrix renormalization-group
(tDMRG) technique, we analyze the propagation of charge, spin and entanglement
in the chains after the quench and relate the light-cone velocities to the
dispersion of holons and spinons. We find that the local magnetization at the
impurity site decays faster if we increase the interaction in the chains, even
though the spin velocity decreases. We derive an analytical expression for the
relaxation of the impurity magnetization which is in good agreement with the
tDMRG results at intermediate timescales, providing valuable insight into the
time evolution of the Kondo screening cloud in interacting systems.
