  In this work we generalise various recent results on the evolution and
monotonicity of the eigenvalues of certain geometric operators under specified
geometric flows. Given a closed, compact Riemannian manifold
$\big(M^n,g(t)\big)$ and a smooth function $\eta\in C^{\infty}(M)$ we consider
the family of operators $\mathbb{L}=\Delta - g(\nabla\eta,\nabla\cdot)+cR$,
where $R$ is the scalar curvature and $c$ is some real constant. We define a
geometric flow on $M$ which encompasses the Ricci, the Ricci - Bourguignon and
the Yamabe flows. Supposing that the metric $g(t)$ evolves along this general
geometric flow we derive a formula for the evolution of the eigenvalues of
$-\mathbb{L}$ and prove monotonicity results for the eigenvalues of both
$-\Delta + g(\nabla\eta,\nabla\cdot)$ and $-\mathbb{L}$. We then prove
Reilly-type formula for the operator $\mathbb{L}$ and employ it to establish an
upper bound for the first variation of the eigenvalues of $-\mathbb{L}$.
Finally, in the pursuit of a theoretical explanation of our generalisations, we
formulate two conjectures on the monotonicity of the eigenvalues of
Schr\"{o}dinger operators.
