  We study the crossover between the sudden quench limit and the adiabatic
dynamics of superconducting states in the attractive Hubbard model. We focus on
the dynamics induced by the change of the attractive interaction during a
finite ramp time which is varied in order to track the evolution of the
dynamical phase diagram from the sudden quench to the equilibrium limit. Two
different dynamical regimes are realized for quenches towards weak and strong
coupling interactions. At weak coupling the dynamics depends only on the energy
injected into the system, whereas a dynamics retaining memory of the initial
state takes place at strong coupling. We show that this is related to a sharp
transition between a weak and a strong coupling quench dynamical regime, which
defines the boundaries beyond which a dynamics independent from the initial
state is recovered. Comparing the dynamics in the superconducting and
non-superconducting phases we argue that this is due to the lack of an
adiabatic connection to the equilibrium ground state for non-equilibrium
superconducting states in the strong coupling quench regime.
