Does the Chapman - Kolmogorov equation hold for an arbitrary stochastic process?

The current wikipedia article on "Stochastic process" ( https://en.wikipedia.org/wiki/Stochastic_process ) seems to say that the Chapman-Kolmogorov equation is not valid for an arbitrary stochastic process.

I say "seems to say" since I can't interpret what the article means by "same class" in the passage:

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

Whether the equation is true for an arbitrary process depends on what one calls "the Chapman-Kolmogorov Equation". In "Handbook Of Stochastic Methods" by C.W. Gardiner, second edition, pages 43-44, marginalization is a proof for the equation: