Let the one-step transition dynamics of a system be given as \begin{equation*} \mathbf{P}[X_{t+1} | X_t] = N(f_\theta(X_t), I) \end{equation*} that is the Normal with mean a function of $X_t$ paramterized by $\theta$. My question is what would be $$\mathbf{P}[X_{t+2} | X_t] = \int_{-\infty}^\infty\mathbf{P}[X_{t+2} | X_{t+1}]\mathbf{P}[X_{t+1} | X_t]dX_{t+1}?$$
Is there a simple closed form solution for a general $f_\theta$? That is would $\mathbf{P}[X_{t+2} | X_t]$ be a Normal Distribution? What if the one step transition is given as a linear function e.g,
\begin{equation*} \mathbf{P}[X_{t+1} | X_t] = N(AX_t+b, I) ? \end{equation*} What would the mean and variance be if it is a normal distribution?