  We uncover an aspect of the Kibble--Zurek phenomenology, according to which
the spectrum of critical exponents of a classical or quantum phase transition
is revealed, by driving the system slowly in directions parallel to the phase
boundary. This result is obtained in a renormalization group formulation of the
Kibble--Zurek scenario, and based on a connection between the breaking of
adiabaticity and the exiting of the critical domain via new relevant directions
induced by the slow drive. The mechanism does not require fine tuning, in the
sense that scaling originating from irrelevant operators is observable in an
extensive regime of drive parameters. Therefore, it should be observable in
quantum simulators or dynamically tunable condensed-matter platforms.
