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).
Best Answer
To address a question asked by the OP in a comment: yes, the LLN yields almost sure results (no expectations here) which show that there exist disjoint sets of complete trajectories such that each Poisson distribution "sees" only one of them (namely $Q_\lambda$ "sees" only the set $D_\lambda$ and $Q_\lambda(D_\mu)=0$ if $\lambda\ne\mu$).
About diffusions with different drifts: as you know, if $(W_t)_{t\ge0}$ is a Brownian motion under $P$ and if $X_t=W_t+\mu t$, then for every $T$, $(X_t)_{0\le t\le T}$ is a Brownian motion (without drift) under a measure $Q_T$ on the space of paths indexed by $[0,T]$. Furthermore, $Q_T$ is absolutely continuous with respect to the restriction $P_T$ of $P$ to the space of paths indexed by $[0,T]$ and the density $dQ_T/dP_T$ is $Z_T=\exp(-\mu W_T-\frac12\mu^2T)$ (this is Cameron-Martin-Girsanov theorem). Thus there exists what you call an equivalent change of measure between $P_T$ and $Q_T$.
But this does not imply that $P$ and $Q$ are mutually absolutely continuous and in fact, they are mutually singular. Note that when $T\to+\infty$, the $P$-martingale $(Z_T)$ converges almost surely when $T\to+\infty$ to $Z_\infty=0$... which is not a Radon-Nykodym density. A more direct proof that $P$ and $Q$ are mutually singular is as follows: for every $\mu$, let $L_\mu$ denote the set of continuous functions $f$ defined on $[0,+\infty[$ such that $f(t)/t\to\mu$ when $t\to+\infty$. If $\mu\ne0$, then $L_0$ and $L_\mu$ are disjoint and $P(X\in L_\mu)=Q(X\in L_0)=1$.