The primary observables in QFT are defined by some local operator-correlation functions. These are ##N##-point functions, which in principle can be calculated for finite times for a given initial state (statistical operator) using the Schwinger-Keldysh real-time contour technique.
Of course, there's the usual trouble with the UV divergences when calculating these N-point functions, and to my knowledge there's not solution for this renormalization problem for such "off-equilibrium situations". You also have to use some resummation since in contradistinction to the special cases of vacuum QFT for calculating S-matrix elements for scattering processes or equilibrium QFT the strictly order-by-order in hbar or coupling constant expansion-parameter perturbation theory is not described by naive perturbation theory.
One approach is the Kadanoff-Baym ##\Phi##-functional method, also known as the two-particle irreducible (2PI) formulation or the Cornwall-Jackiw-Tomboulis approach. There you derive self-consistent approximations for the one-body Green's function and the corresponding self-consistent self-energy, which is given by "skeleton diagrams" derived from the functional approach. Despite the notorious (and to my knowledge unsolved) renormalization problem there are a view studies solving the full Kadanoff-Baym equations for simple truncations and in lower space-time dimensions (to soften the renormalizability problem). One example is a study in simple ##\phi^4## theory in (1+2) dimensions
https://arxiv.org/abs/nucl-th/0401046
https://doi.org/10.1016/j.nuclphysa.2004.07.010
Usually the Kadanoff-Baym equations are used to derive quantum-transport equations, using additional approximations like the gradient expansion ("coarse graining"), leading to transport theories that work with some caveats even for broad resonances rather than "particles". It's further simplified if the spectral functions don't develop too large (collisional) widths. Then you can also apply the quasiparticle approximation, which leads to usual relativistic Boltzmann-Uehling-Uhlenbeck transport equations. For this there's a vast amount of literature related with the study of ultrarelativistic heavy-ion collisions. Some examples are
Y. B. Ivanov, J. Knoll, H. v. Hees and D. N. Voskresensky,
Soft Modes, Resonances and Quantum Transport, Phys.
Atom. Nucl. 64, 652 (2001),
https://arxiv.org/abs/nucl-th/0005075Y. B. Ivanov, J. Knoll and D. N. Voskresensky, Self-consistent
approximations to non-equilibrium many-body theory, Nucl.
Phys. A 657, 413 (1999),
https://arxiv.org/abs/hep-ph/9807351
Y. B. Ivanov, J. Knoll and D. Voskresensky, Resonance
transport and kinetic entropy, Nucl. Phys. A 672, 313 (2000),
https://doi.org/10.1016/S0375-9474(99)00559-X
Y. B. Ivanov, J. Knoll and D. N. Voskresensky, Resonance
Transport and Kinetic Entropy, Nucl. Phys. A 672, 313
(2000),
https://arxiv.org/abs/nucl-th/9905028
Y. Ivanov, J. Knoll and D. Voskresensky, Selfconsistent
approach to off-shell transport, Phys. Atom. Nucl. 66, 1902
(2003),
https://doi.org/10.1134/1.1619502
J. Knoll, Y. B. Ivanov and D. Voskresensky, Exact
Conservation Laws for the Gradient Expanded
Kadanoff-Baym Equations, Ann. Phys. (NY) 293, 126 (2001),
https://arxiv.org/abs/nucl-th/0102044
W. Cassing, From Kadanoff-Baym dynamics to off-shell
parton transport, Eur. Phys. J. ST 168, 3 (2009),
https://doi.org/10.1140/epjst/e2009-00959-x
and many more.