I've seen a definition of Markov chains as a stochastic process $(X_t)_{t\in I}$ fulfilling the weak Markov property and having index set $I = \mathbb{N}_0$.
But the weak Markov property $$\mathbf{P}_x[X_{t+s} \in A \mid \mathcal{F}_s] = \kappa_t(X_s, A) \quad \mathbf{P}_x-\text{a.s.}$$
seems to entail time-homogenity.
So is it correct that a process like $(Z_t)_{t\in \mathbb{N_0}}$, where $Z_t$ is for $0 \leq t \leq 100$ a centered symmetric random walk in $\mathbb{Z}$ and then $Z_t = 0$ for $t \geq 101$, is not a Markov chain, though the following holds: $$\mathbf{P}[Z_t = z \mid Z_{s_1} = z_1, \ldots, Z_{s_n} = z_n] = \mathbf{P}[Z_t = z \mid Z_{s_n} = z_n]\; \text{ with } \; s_1 < \cdots < s_n < t \, .$$
Time homogeneity is not a requirement of Markov Chains. All that is needed is that the Markov Chain be memoryless. The conditional probability distribution of future values of the stochastic process given the current and past states, must be equal to the conditional probability distribution of future values of the stochastic process given the current state.
Transition probabilities of Markov Chains most definitely can depend on time. The ones that don't are called time-homogeneous.
For instance in a discrete time discrete state Markov Chain, rather than having a single transition matrix P for each of the transition epochs, you could have P1, P2, P3, say for a 3 period Markov Chain. The transition matrix over the 3 periods = P3 * P2 * P1, as opposed to P^3 if they are all equal to P. The same idea applies to continuous time and/or continuous state.
See also https://mathoverflow.net/questions/168398/time-inhomogeneous-markov-chains