  Eclipsing binary systems with components that pulsate in gravity modes ($g$
modes) allow for simultaneous and independent constraints of the chemical
mixing profiles of stars. We aim to assemble a sample of $g$-mode pulsators in
detached eclipsing binaries with the purpose of finding good candidates for
future evolutionary and asteroseismic modelling. In addition, we present a case
study of the eclipsing binary KIC9850387, identified as our most promising
candidate. We selected all of the detached eclipsing binaries in the Kepler
eclipsing binary catalogue and performed a visual inspection to determine the
presence and density of $g$ modes, and the presence of $g$-mode period-spacing
patterns in their frequency spectra. We then characterised our sample based on
their $g$-mode pulsational parameters and binary and atmospheric parameters. A
spectroscopic follow-up of KIC9850387 was then performed. We then performed
spectral disentangling followed by atmospheric modelling and abundance analysis
for the primary star. We utilised an iterative approach to simultaneously
optimise the pulsational and eclipse models, and subsequently performed an
analysis of the pressure- ($p$-) and $g$-mode pulsational frequencies. We
compiled a sample of 93 Kepler eclipsing binary stars with $g$-mode pulsating
components and identified clear $g$-mode period-spacing patterns in the
frequency spectra of seven of these systems. We found that the $g$-mode
pulsational parameters and the binary and atmospheric parameters of our sample
are weakly correlated at best. We find that the eclipsing binary KIC9850387 is
a double-lined spectroscopic binary in a near-circular orbit with an
intermediate-mass primary, and a solar-like secondary. We find $\ell=1$ and
$\ell=2$ period-spacing patterns in the frequency spectrum of KIC9850387
spanning more than ten radial orders each.
