Let $\Omega=\{f:\mathbb R^+\to \mathbb R| \varphi(f)\}$, where $\varphi$ is some useful statement about functions. For example $\varphi(f)$ could mean that $f$ is continuous. Endow $\Omega$ with the $\sigma$-algebra $\mathcal F=\Omega\cap\mathcal B(\mathbb R)^{\mathbb R^+}$ and an arbitrary probability measure. Let $X$ be the canonical process on $\Omega$. By that I mean $X_t(f)=f(t)$. Suppose further that $\tau_X$ is a $(\mathcal F_t^X)_t$ stopping time that depends on $X$. By this I means that in the definition of $\tau_X$ we could replace each instance of $X$ with another process $Y$ defined on some other probability space to form $\tau_Y$, and it would remain well defined. The natural examples would be hitting and exit times.
Now consider a filtered probability space $(\Omega',\mathcal F',(\mathcal F'_t)_t,\mathbb P)$. Let $Y$ be a stochastic process on this space such that the function $t\mapsto Y_t(\omega)$ satisfies $\varphi$ for all $\omega\in \Omega'$. Can we say that $\tau_Y$ is a stopping time because $\tau_X$ is a stopping time? I am fairly certain the answer is no, for although $Y:\Omega'\to \Omega$ given by $(Y(\omega))_t=Y_t(\omega)$ is a bijection and can push forward the measure to maintain the law, I see no reason why it should preserve the structure of the filtration or $\sigma$-algebra.
Are there any reasonable restrictions we could make so that such an implication holds? For example insisting that $(\mathcal F_t')_t=(\mathcal F_t^Y)_t$? It would be useful if such restrictions did exist, because then in these cases we could use the Galmarino test to determine whether a random variable on a space is a stopping time.