  In this paper, we present a general derivation of a modified
fluctuation-dissipation theorem (MFDT) valid near an arbitrary non-stationary
state for a system obeying markovian dynamics. We show that the method to
derive modified fluctuation-dissipation theorems near non-equilibrium
stationary states used by J. Prost et al., PRL 103, 090601 (2009), is
generalizable to non-stationary states. This result follows from both standard
linear response theory and from a transient fluctuation theorem, analogous to
the Hatano-Sasa relation. We show that this modified fluctuation-dissipation
theorem can be interpreted at the trajectory level using the notion of
stochastic trajectory entropy, in a way which is similar to what has been done
recently in the case of MFDT near non-equilibrium steady states (NESS). We
illustrate this framework with two solvable examples: the first example
corresponds to a brownian particle in an harmonic trap submitted to a quench of
temperature and to a time-dependent stiffness. The second example is a classic
model of coarsening systems, namely the 1D Ising model with Glauber dynamics.
