  Heavy ion reactions and other collective dynamical processes are frequently
described by different theoretical approaches for the different stages of the
process, like initial equilibration stage, intermediate locally equilibrated
fluid dynamical stage and final freeze-out stage. For the last stage the best
known is the Cooper-Frye description used to generate the phase space
distribution of emitted, non-interacting, particles from a fluid dynamical
expansion/explosion, assuming a final ideal gas distribution, or (less
frequently) an out of equilibrium distribution. In this work we do not want to
replace the Cooper-Frye description, rather clarify the ways how to use it and
how to choose the parameters of the distribution, eventually how to choose the
form of the phase space distribution used in the Cooper-Frye formula. Moreover,
the Cooper-Frye formula is used in connection with the freeze-out problem,
while the discussion of transition between different stages of the collision is
applicable to other transitions also. More recently hadronization and molecular
dynamics models are matched to the end of a fluid dynamical stage to describe
hadronization and freeze-out. The stages of the model description can be
matched to each other on spacetime hypersurfaces (just like through the
frequently used freeze-out hypersurface). This work presents a generalized
description of how to match the stages of the description of a reaction to each
other, extending the methodology used at freeze-out, in simple covariant form
which is easily applicable in its simplest version for most applications.
