= Solution
The gradient bound makes $f$ a <Lipschitz function>, with $|f(x)|\le|f(0)|+K\lVert x\rVert$, so its value at a <Gaussian random variable> is integrable and square integrable. The future <Brownian increment> $X_1-X_t$ is independent of $\mathcal F_t$ and has distribution $N(0,(1-t)I_d)$. Conditioning on the current location therefore gives
$$
\boxed{M_t=\mathbb E[f(X_t+(X_1-X_t))\mid\mathcal F_t]=P_{1-t}f(X_t).}
$$
This is the <Markov property> expressed through the <Brownian transition semigroup>. In particular $M_0=\mu$ and $M_1=f(X_1)$.
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