  We study the formation of magnon-polaron excitations and the consequences of
different time scales between the magnon and lattice dynamics. The spin-spin
interactions along the 1D lattice are ruled by a Heisenberg Hamiltonian in the
anisotropic form XXZ, in which each spin exhibits a vibrational degree of
freedom around its equilibrium position. By considering a magnetoelastic
coupling as a linear function of the relative displacement between
nearest-neighbor spins, results provide an original framework for achieving a
hybridized state of magnon-polaron. Such state is characterized by high
cooperation between the underlying excitations, where the traveling or
stationary formation of magnon-polaron depends on the effective magnetoelastic
coupling. A systematic investigation reveals the critical amount of the
magnon-lattice interaction ($\chi_c$) necessary to emergence of the stationary
magnon-polaron quasi-particle. Different characteristic time scales of the
magnon and the vibrational dynamics unveiled the threshold between the two
regimes, as well as a limiting value of critical magnetoelastic interaction,
above which the magnon velocity no longer interferes at the critical
magnetoelastic coupling capable of inducing the stationary regime.
