  We study the evolution of dynamic properties of the BCS/BEC (Bose-Einstein
Condensate) crossover in a relativistic superfluid as well as its
thermodynamics. We put particular focus on the change in the soft mode dynamics
throughout the crossover, and find that three different effective theories
describe it; these are, the time-dependent Ginzburg-Landau (TDGL) theory in the
BCS regime, the Gross-Pitaevskii (GP) theory in the BEC regime, and the
relativistic Gross-Pitaevskii (RGP) equation in the relativistic BEC (RBEC)
regime. Based on these effective theories, we discuss how the physical nature
of soft mode changes in the crossover. We also discuss some fluid-dynamic
aspects of the crossover using these effective theories with particular focus
on the shear viscosity. In addition to the study of soft modes, we show that
the ``quantum fluctuation'' is present in the relativistic fermion system,
which is in contrast to the usual Nozi`eres--Schmit-Rink (NSR) theory. We
clarify the physical meaning of the quantum fluctuation, and find that it
drastically increases the critical temperature in the weak coupling BCS regime.
