  We consider evolutionary PDE inclusions of the form \[
  -\lambda \dot{u}_\lambda + \Delta u - \mathrm{D} W_0(u) + f \ni \partial
\mathcal{R}_1(\dot{u}) \qquad\text{in $(0,T) \times \Omega$,} \] where
$\mathcal{R}_1$ is a positively $1$-homogeneous rate-independent dissipation
potential and $W_0$ is a (generally) non-convex energy density. This work
constructs solutions to the above system in the slow-loading limit $\lambda
\downarrow 0$. Our solutions have more regularity both in space and time than
those that have been obtained with other approaches. On the "slow" time scale
we see strong solutions to a purely rate-independent evolution. Over the jumps,
we obtain a detailed description of the behavior of the solution and we resolve
the jump transients at a "fast" time scale, where the original rate-dependent
evolution is still visible. Crucially, every jump transient splits into a
(possibly countable) number of rate-dependent evolutions, for which the energy
dissipation can be explicitly computed. This in particular yields a global
energy equality for the whole evolution process. It also turns out that there
is a canonical slow time scale that avoids intermediate-scale effects, where
movement occurs in a mixed rate-dependent / rate-independent way. In this way,
we obtain precise information on the impact of the approximation on the
constructed solution. Our results are illustrated by examples, which elucidate
the effects that can occur.
