Dynamic critical behavior in superfluid systems is considered in the presence of external stirring and advecting processes. The latter are generated by means of the Gaussian random velocity ensemble with white-noise character in time variable and self-similar spatial dependence. The main focus of this work is to analyze an effect of compressible modes on the critical behavior. The model is formulated through stochastic Langevin equations, which are then recast into the Janssen-De Dominicis response formalism. Employing the field-theoretic perturbative renormalization group method we analyze large-scale properties of the model. Explicit calculations are performed to the leading one-loop approximation in the double (ɛ,y) expansion scheme, where ɛ is a deviation from the upper critical dimension d_{c}=4 and y describes a scaling property of the velocity ensemble. Altogether five distinct universality classes are expected to be macroscopically observable. In contrast to the incompressible case, we find that compressibility leads to an enhancement and stabilization of nontrivial asymptotic regimes.