You can directly deal with inhomogeneous Markov processes through the Kolmogorov backward and forward equations. I suppose you are asking for the forward equation (i.e. derivative with respect to the time in the future). Let me discuss things on a formal level, which means here I ignore regularity conditions to ensure the existence of generators, derivatives, etc etc.

Let me first give the backward equation, which is probably easier (a specific case of Feynman-Kac formula). Let $u(s,x,t)=\mathbb{E}^{s,x} f(X_t)$ for a nice function $f$ (e.g. infinitely differentiable with compact support), where the expectation is under the measure $\mathbb{P}^{s,x}$ such that $\mathbb{P}^{s,x}\{X_s=x\}=1$. Let $A_s$ be the generator (I am ignoring the detail of how $A_s$ is defined, you can probably figure it out yourself or see references I give below). Then the backward equation states that

$\frac{\partial u}{\partial s}+A_s u =0$.

The forward equation is usually formulated for the density $p(s,x,t,y)$ of the transition kernel $P_{s,t}(x,A)=\int p(s,x,t,y)dy$:

$\frac{\partial p}{\partial t}=A_t^* p(s,x,t,y)$

where $A_t^*$ is the adjoint of the operator $A_t$ (again, there's subtlety in how you definite $A_t$ for forward and backward equations separately, I ignore that here).

For a given diffusion given as an SDE $dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$ for a Wiener process $W$, it is pretty straightforward that $A_t f(x)=\mu(t,x)f'(x)+\frac{1}{2}\sigma^2(t,x)f''(x)$.

All these are discussed to certain extent in "The Theory of Stochastic Processes, Vol II" by Gikhman and Skorokhod, and "Multidimensional Diffusion Processes" by Stroock and Varadhan (expressed in an integral form instead of derivatives).

You can directly deal with inhomogeneous Markov processes through the Kolmogorov backward and forward equations. I suppose you are asking for the forward equation (i.e. derivative with respect to the time in the future). Let me discuss things on a formal level, which means here I ignore regularity conditions to ensure the existence of generators, derivatives, etc etc.

Let me first give the backward equation, which is probably easier (a specific case of Feynman-Kac formula). Let $u(s,x,t)=\mathbb{E}^{s,x} f(X_t)$ for a nice function $f$ (e.g. infinitely differentiable with compact support), where the expectation is under the measure $\mathbb{P}^{s,x}$ such that $\mathbb{P}^{s,x}\{X_s=x\}=1$. Let $A_s$ be the generator (I am ignoring the detail of how $A_s$ is defined, you can probably figure it out yourself or see references I give below). Then the backward equation states that

$\frac{\partial u}{\partial s}+A_s u =0$.

The forward equation is usually formulated for the density $p(s,x,t,y)$ of the transition kernel $P_{s,t}(x,B)=\int_B p(s,x,t,y)dy$:

$\frac{\partial p}{\partial t}=A_t^* p(s,x,t,y)$

where $A_t^*$ is the adjoint of the operator $A_t$ (again, there's subtlety in how you definite $A_t$ for forward and backward equations separately, I ignore that here).

For a given diffusion given as an SDE $dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$ for a Wiener process $W$, it is pretty straightforward that $A_t f(x)=\mu(t,x)f'(x)+\frac{1}{2}\sigma^2(t,x)f''(x)$.

All these are discussed to certain extent in "The Theory of Stochastic Processes, Vol II" by Gikhman and Skorokhod, and "Multidimensional Diffusion Processes" by Stroock and Varadhan (expressed in an integral form instead of derivatives).