We give an example to show that, given a nonhomogeneous Markov process $X$ on $E$, one cannot, in general, produce a right continuous strong Markov version of $X$ on a compactification of $E$. In particular, the space-time regularization fails to produce one compact Hausdorff state space for the nonhomogeneous process.
"Note on the Ray-Knight Compactification." Ann. Probab. 7 (3) 543 - 546, June, 1979. https://doi.org/10.1214/aop/1176995055