We present a dynamical extension of the phonon-induced spin-crossover (SC) transition model, which allows obtaining the time evolution of the metastable high spin fraction at low temperature. The model is based on 1D lattices of two level atoms coupled by springs. This model is somehow isomorphic with an Ising-like model in which the interactions between the spins take place through lattice distortions. The thermodynamic properties are studied in the frame of a self-consistent phonon approach, which gives an elegant variational formalism in which the 1D character of the system is preserved. Obviously, this kind of mean-field approach allows the existence of phase transitions. Thus, we found that first-order and two-step SC transitions can be obtained, depending on the choice of the elastic constants between the atoms. The dynamical version of this model is also investigated by using the master equation formalism. The equations of motion of the high-spin (HS) fraction and equal-time two-spin correlation are derived analytically in the frame of a local equilibrium approach, in which we consider that the phonon bath equilibrates instantaneously, in contrast to the dynamics of the macrovariables such as the high-spin fraction. Sigmoidal curves with long relaxation tails are obtained for the time dependence of the HS fraction in the ferroelastic case. The flow diagrams and the time dependence of the configuration entropy upon relaxation are analyzed, and the obtained results are discussed in relation to those of the dynamical mean-field theory, which is the approach most used to describe the experimental results, although it neglects short-range fluctuations.