My textbook claims the following fact. As far as I can tell it provides neither a proof nor a citation.
Fact: Let $S\subseteq\mathbb Z^d$ be infinite. Let $\zeta_1,\cdots,\zeta_n\in\mathbb Z^d$ be distinct. For $\ell=1,\cdots,n$, let $\lambda_\ell:S\to\mathbb R_{\ge0}$. Suppose that $\{X(t)\}_{t\ge0}$ is a continuous time Markov chain with state space $S$ and transition rate matrix $Q=(q_{xy})_{xy\in S}$ where, for $x\ne y$,
$\begin{equation} q_{xy}=\begin{cases} \lambda_\ell(x) & \zeta_\ell=y-x\\ 0 & \forall\ell, \zeta_\ell\ne y-x \end{cases}. \end{equation}$
Then there exists iid unit rate Poisson processes $Y_1,\cdots,Y_n$ such that
$\begin{equation}\displaystyle X(t)=X(0)+\sum_{\ell=1}^n Y_\ell\left(\int_0^t\lambda_\ell(X(s))\,\mathrm ds\right)\zeta_\ell. \end{equation}$
Thoughts
The intuition is clear enough. $\lambda_\ell(X(s))$ is the intensity of the transition in the direction $\zeta_\ell$ at time $s$, so the expected number of such transitions up to time $t$ is $\int_0^t\lambda_\ell(X(s))\,\mathrm ds$. The actual number of these transitions as a function of $t$ should be some sort of point process, and Poisson point processes are the Gaussian random variables of point processes, so it's believable that the number of transitions up to time $t$ should take the form $Y_\ell\left(\int_0^t\lambda_\ell(X(s))\,\mathrm ds\right)$. Multiplying by $\zeta_\ell$ we get the total change contributed by transitioning in the direction $\zeta_\ell$, and so the sum should be the total change period. The tricky part, I suppose, is showing that the transitions in direction $\zeta_\ell$ really are such a Poisson process.
My guess is that this is supposed to be an obvious consequence of the preceding material in the book, but I don't see how it follows. Either proving the fact or giving a textbook that does would be an acceptable answer.