It is well-known that, for a continuous-time Markov process, the transition probabilities $\mathbf{P} = \mathbb{P}\left[X(t) = j |X(0) = i\right]$ satisfy the following evolution equation:
$$\frac{d\mathbf{P}}{dt} = \mathbf{A}{\mathbf{P}}$$
where $\mathbf{A}$ is a time-independent matrix usually called the generator matrix.
If I were to identify a stochastic process whose transition probabilities evolve according to this equation, would this immediately imply the process possesses the Markov property?