  Recently a modified version of Rastall theory of gravity has been introduced
in which a varying coupling parameter could act as dark energy (DE) and thus,
it can be held responsible for the current accelerated expansion of the
Universe. Motivated by this modification, we study here the evolution of linear
and non-linear perturbations in the matter content of the Universe, utilizing
spherically symmetric top-hat collapse scenario. The exact solutions we obtain
in linear regime show that as the Universe evolves, matter density
perturbations grow and reach a maximum value at a certain redshift after which
these perturbations start decreasing towards a finite positive value at the
present time. Depending on model parameters, exact oscillatory solutions can be
also found representing that matter perturbations could experience either
overdense and underdense regions during the dynamical evolution of the
Universe. Numerical solutions in non-linear regime show that the amplitude of
perturbations grow much faster than the linear one and diverges at a critical
redshift. However, the formation of collapsed structures is delayed as compared
to $\Lambda${\rm CDM} model. It is found that the running mutual interaction
between matter and geometry, encoded in the variable Rastall coupling
parameter, could drastically affect the dynamics of matter perturbations and
their growth rate during the evolution of the Universe.
