Definition: $$h(x)=f*g(x)=\int_A f(x-t)g(t)dt$$ where A is a support of function $q()$, i.e. $A=\{t:q(t)\ne 0\}$

Let's calculate derivative:

$$\frac {dh}{dx}=\underset{dx\rightarrow0}{\lim} \frac {(\int_A f(x+dx-t)g(t)dt-\int_A f(x-t)g(t)dt)}{dx}=\underset{dx\rightarrow0}{\lim}(\int_A \frac{(f(x+dx-t)-f(x-t))}{dx}g(t)dt)$$

If we assume that there exists some integrable function $q(t)$, such that for $t$ almost everywhere
$$
\left| \frac{(f(x+dx-t)-f(x-t))}{dx} \right| < q(t), \forall dx>0
$$

I.e.
$$
\mu\{t: \left| \frac{(f(x+dx-t)-f(x-t))}{dx} \right| \ge q(t)\}=0,\forall dx >0
$$

then by the Lebesgue dominated convergence theorem we can push the limit inside integral.

$$\frac {dh}{dx}=\frac{d}{dx}(f*g(x))=\int_A f'(x-t)g(t)dt=f'*g$$

Under assumption that: $\int_A q(t)dt$ is bounded above.
One situation is when A is a compact set and $f,g$ are continuous function in the set A with a finite number of dicontinuities.