Schwinger's quantum action principle states that,
\begin{equation}
\delta\langle x_{2},t_{2}|x_{1},t_{1}\rangle=i\langle x_{2},t_{2}|\delta\hat{S}|x_{1},t_{1}\rangle
\end{equation}
where $|x,t\rangle$ is an eigenstate of the time dependent Heisenberg picture position operator and $\hat{S}$ is the action promoted to be an operator.
Schwinger argues that the variation of the amplitude $\langle x_{2},t_{2}|x_{1},t_{1}\rangle$ can only depend on the initial and final states since these are the only objects making the amplitude. The variation in the amplitude must be,
\begin{equation}
\delta\langle x_{2},t_{2}|x_{1},t_{1}\rangle=\delta (\langle x_{2},t_{2}|)|x_{1},t_{1}\rangle + \langle x_{2},t_{2}|\delta(|x_{1},t_{1}\rangle) \ .
\end{equation}
The variation of the eigenstates is assumed to be the result of a small unitary operator,
\begin{equation}
|x,t\rangle+\delta |x,t\rangle=e^{-i\hat{G}}|x,t\rangle
=|x,t\rangle-i\hat{G}|x,t\rangle
\end{equation}
so that,
\begin{equation}
\delta |x,t\rangle=-i\hat{G}|x,t\rangle \ .
\end{equation}
The variation of the amplitude is now,
\begin{equation}
\delta\langle x_{2},t_{2}|x_{1},t_{1}\rangle=i\langle x_{2},t_{2}|\hat{G}_{2}|x_{1},t_{1}\rangle -i \langle x_{2},t_{2}|\hat{G}_{1}|x_{1},t_{1}\rangle
=i\langle x_{2},t_{2}|(\hat{G}_{2}-\hat{G}_{1})|x_{1},t_{1}\rangle \ .
\end{equation}
Comparing this result with the statement of Schwinger's quantum action principle implies,
\begin{equation}
\delta\hat{S}=\delta\int_{t_{1}}^{t_{2}}\hat{L}dt=\hat{G}_{2}-\hat{G}_{1} \ .
\end{equation}
This relation is saying that the variation of the action operator only involves dynamical variables at the terminal times (Quantum Kinematics and Dynamics, page 78) and that over the interior of the duration the action operator is a constant operator.
As an example, suppose we vary the end time from $t_{2}$ to $t_{2}+\delta t_{2}$. In classical mechanics, the variation of the action as a function of the coordinates of the start and end of the path is,
\begin{equation}
dS=p_{2}dx_{2}-H_{2}dt_{2}-p_{1}dx_{1}+H_{1}dt_{1}
\end{equation}
so that varying the end time and promoting the result to an operator gives $\delta \hat{S}=-\hat{H}_{2}\delta t_{2}$. Schwinger's quantum action principle now gives Schrodinger's equation in the form,
\begin{equation}
\delta\langle x_{2},t_{2}|x_{1},t_{1}\rangle=-i\langle x_{2},t_{2}|\hat{H}_{2}|x_{1},t_{1}\rangle\delta t_{2} \ .
\end{equation}
