  We examine how the binding of light ($A\leq 8$) nuclei depends on possible
variations of hadronic masses, including meson, nucleon, and nucleon-resonance
masses. Small variations in hadronic masses may have occurred over time; the
present results can help evaluate the consequences for big bang
nucleosynthesis. Larger variations may be relevant to current attempts to
extrapolate properties of nucleon-nucleon interactions from lattice QCD
calculations. Results are presented as derivatives of the energy with respect
to the different masses so they can be combined with different predictions of
the hadronic mass-dependence on the underlying current-quark mass $m_q$. As an
example, we employ a particular set of relations obtained from a study of
hadron masses and sigma terms based on Dyson-Schwinger equations and a
Poincar\'{e}-covariant Faddeev equation for confined quarks and diquarks. We
find that nuclear binding decreases moderately rapidly as the quark mass
increases, with the deuteron becoming unbound when the pion mass is increased
by $\sim$60% (corresponding to an increase in $X_q=m_q/\Lambda_{QCD}$ of 2.5).
In the other direction, the dineutron becomes bound if the pion mass is
decreased by $\sim$15% (corresponding to a reduction of $X_q$ by $\sim$30%). If
we interpret the disagreement between big bang nucleosynthesis calculations and
measurements to be the result of variation in $X_q$, we obtain an estimate
$\delta X_q/X_q=K \cdot (0.013 \pm 0.002)$ where $K \sim 1$ (the expected
accuracy in $K$ is about a factor of 2). The result is dominated by $^7$Li
data.
