I'll try to ask this question in the most simplistic environment possible, so lets think about a household that can consume and save. After some cumbersome work, we have solved for the optimal savings policy, and know that the change of assets, given his assets $a$, is given by $s(a) = \dot a(a)$.
Denoting with $g(a, t)$ the density of households with assets at level $a$ given time $t$, I can compute the Kolmogorov Forward Equation for the distribution of assets:
$$ \partial_t g(a, t) = - \partial_a [s(a,t)g(a,t)]$$
Where I followed the methodology of Achdou et al, Appendix A3: First compute the CDF $G(a, t)$, and then use that $g(a,t) = \partial_t G(a,t)$.
However, now assume that we have birth and death of households. Most importantly, I don't want there to be a constant death rate, but let's make the death rate conditional on the asset level: $d(a,t)$ - and let's denote birth by $b(a,t)$. Intuitively, I'd think that the new KFE would be given by
$$ \partial_t g(a, t) = - \partial_a [s(a,t)g(a,t)] - d(a,t)g(a,t) + b(a,t)$$
Intuition aside, how can I derive the correct KFE that allows for death rates and birth from the scratch?