Can I use Leibniz' rule to differentiate when integral is $\int_0^\gamma f(\gamma,s)dF(s)$, instead of $\int_0^\gamma f(\gamma,s)ds$ ($F$ is a CDF)

I will interpret the integral as Riemann-Stieltjes integral. Then by integration by parts,

\begin{align*} \int_{0}^{\gamma} (\gamma-s) \, dF(s) &= \left[ (\gamma-s)F(s) \right]_{0}^{\gamma} + \int_{0}^{\gamma} F(s) \, ds \\ &= -\gamma F(0) + \int_{0}^{\gamma} F(s) \, ds \\ &= \int_{0}^{\gamma} (F(s) - F(0)) \, ds. \end{align*}

So we have

$$ \frac{d}{d\gamma} \int_{0}^{\gamma} (\gamma-s) \, dF(s) = F(\gamma) - F(0) = \int_{0}^{\gamma} dF(s) $$

at every continuity point $\gamma$ of $F$. But also notice that this is exactly what we expect when applying the Leibniz integral rule:

$$ \frac{d}{d\gamma} \int_{0}^{\gamma} (\gamma-s) \, dF(s) \quad``\,=\text{''}\quad \underbrace{(\gamma - \gamma)F'(\gamma)}_{=0} + \int_{0}^{\gamma} dF(s). $$