Laplace transform of $\int_0^t f(s,t) ds$ I know how to find the Laplace transform of an integral of the form $\Gamma(t) := \int_0^t f(s) ds$, this is simply applying a formula, but I was wondering if a formula for the more general $\Gamma(t) := \int_0^t f(s,t) ds$ exists as well.

The particular case I am interested in is $\int_0^t f(s-t) g(s) ds$ for arbitrary functions $f$ and $g$.
Assuming that your complex argument is $s$ :

$$\mathcal{L}\bigg\{\int_0^t f(s-t) g(s) ds\bigg\}(ω) = \frac{\mathcal{L}\{f(0)g(t)\}(ω)}{ω}$$
Assuming that your complex argument is $s$ :

$$\mathcal{L}\bigg\{\int_0^t f(s-t) g(s) ds\bigg\}(ω) = \frac{\mathcal{L}\{f(0)g(t)\}(ω)}{ω}$$