  Efficient simulations of the dynamics of open systems is of wide importance
for quantum science and tech-nology. Here, we introduce a generalization of the
transfer-tensor, or discrete-time memory kernel, formalism to multi-time
measurement scenarios. The transfer-tensor method sets out to compute the state
of an open few-body quantum system at long times, given that only short-time
system trajectories are available. Here, we showthat the transfer-tensor method
can be extended to processes which include multiple interrogations (e.g.
measurements) of the open system dynamics as it evolves, allowing us to
propagate high order short-time correlation functions to later times, without
further recourse to the underlying system-environment evolution. Our approach
exploits the process-tensor description of open quantum processes to represent
and propagate the dynamics in terms of an object from which any multi-time
correlation can be extracted. As an illustration of the utility of the method,
we study the build-up of system-environment correlations in the paradigmatic
spin-boson model, and compute steady-state emission spectra, taking fully into
account system-environment correlations present in the steady state.
