  Within the ultimate goal of classifying universality in quantum many-body
dynamics, understanding the relation between out-of-equilibrium and equilibrium
criticality is a crucial objective. Models with power-law interactions exhibit
rich well-understood critical behavior in equilibrium, but the
out-of-equilibrium picture has remained incomplete, despite recent experimental
progress. We construct the rich dynamical phase diagram of free-fermionic
chains with power-law hopping and pairing, and provide analytic and numerical
evidence showing a direct connection between nonanalyticities of the return
rate and zero crossings of the string order parameter. Our results may explain
the experimental observation of so-called \textit{accidental} dynamical
vortices, which appear for quenches within the same topological phase of the
Haldane model, as reported in [Fl\"aschner \textit{et al.}, Nature Physics
\textbf{14}, 265 (2018)]. Our work is readily applicable to modern
ultracold-atom experiments, not least because state-of-the-art quantum gas
microscopes can now reliably measure the string order parameter, which, as we
show, can serve as an indicator of dynamical criticality.
