The S-matrix formalism can be applied to effective field theories involving unstable particles (simply by giving the masses a small imaginary part encoding their half-life - no other change to the usual setting).

Apart from that I do not know anything resembling an asymptotic theory for dissipative systems.

There is however, an interesting connection between S-matrices of QFT and master equations, explained in some detail in Weinberg's Quantum Field Theory book, Vol I, Section 3.6. Indeed, the H-theorem for the Pauli master equation is shown there to be intimately related to the unitarity of the S-matrix, via the Low equations and the resulting optical theorem. (If you are interested in that I could expand a bit on this.)

OK, here are some more details about the latter connection: Weinberg relates the S-matrix elements to transition rates (rather than transition probabilities) in Section 3.4, and uses this relation to define a corresponding master equation; the rate of change of the probability density $P_\alpha$ (where $\alpha$ indexes the possible scattering eigenstates) is obtained by the usual balance equation matching what goes in and what goes out (Weinberg, eq. (3.6.19)). The H-theorem for this equation can then be proved assuming that the transition rates satisfy detailed balance. This detailed balance condition is a consequence of time reversal invariance together with the Low equations. The latter are equivalent to the unitarity of the S-matrix.

The Master equation is meaningful of course only in a context where many collisions take place everywhere, thus in a chemical or nuclear reaction context. Time reversal invariance also requires the absence of external magnetic fields.