Universal behavior is a typical emergent feature of critical systems. A paramount model of the non-equilibrium critical behavior is the directed bond percolation process that exhibits an active- to-absorbing state phase transition in the vicinity of a percolation threshold... Fluctuations of the ambient environment might affect or destroy the universality properties completely. In this work we assume that the random environment can be described by means of compressible velocity fluctu- ations. Using field-theoretic models and renormalization group methods we investigate large-scale and long-time behavior. Altogether eleven universality classes are found, out of which four are stable in the infrared limit and thus macroscopically accessible. In contrast to the model without veloc- ity fluctuations a possible candidate for a realistic three-dimensional case, a regime with relevant short-range noise, is identified. Depending on the dimensionality of space and the structure of the turbulent flow we calculate critical exponents of the directed percolation process. In the limit of the purely transversal velocity field random force critical exponents comply with the incompressible results obtained by previous authors. We have found intriguing non-universal behavior related to the mutual effect of compressibility and advection. (read more)