Let $X_0\sim \mathcal{L}([0,1]^n)$ where $\mathcal{L}([0,1]^n)$ is some continuous distribution over $[0,1]$ and consider a finite set of doubly stochastic invertible $n\times n$ matrices $\mathcal{M}$. Define for $t\in\mathbb{N}$ the process $X_t = \Phi^t X_0$ where $\Phi\sim\mathrm{Unif}(\mathcal{M})$. Conditional on $X_0 = y$ for some fixed $y\in[0,1]^n$ the process can only take discrete values since there are only finitely many matrices (argument would also work for countably many).
How do I interpret probabilities like $\mathbb{P}[X_t = z| X_0 =y]$? Since $X_0$ is a continuous RV the event $\{X_0 = y\}$ has no mass and hence conditioning on it seems wrong. Furthermore, by conditioning on the initial value the whole process becomes discrete while without conditioning it $X_t$ is a random linear transformation (invertible) of $X_0$ and hence also continuous. I am a bit confused how to deal with this role of $X_0$.
Thank you in advance for any help.