  We introduce a general theory on stationary approximations for locally
stationary continuous-time processes. Based on the stationary approximation, we
use $\theta$-weak dependence to establish laws of large numbers and central
limit type results under different observation schemes. Hereditary properties
for a large class of finite and infinite memory transformations show the
flexibility of the developed theory. Sufficient conditions for the existence of
stationary approximations for time-varying L\'evy-driven state space models are
derived and compared to existing results. We conclude with comprehensive
results on the asymptotic behavior of the first and second order localized
sample moments of time-varying L\'evy-driven state space models.
