  A discrete Gibbsian line ensemble $\mathfrak{L} = (L_1,\dots,L_N)$ consists
of $N$ independent random walks on the integers conditioned not to cross one
another, i.e., $L_1 \geq \cdots \geq L_N$. In this paper we provide sufficient
conditions for convergence of a sequence of suitably scaled discrete Gibbsian
line ensembles $f^N = (f_1^N,\dots,f_N^N)$ as the number of curves $N$ tends to
infinity. Assuming log-concavity and a KMT-type coupling for the random walk
jump distribution, we prove that under mild control of the one-point marginals
of the top curves with a global parabolic shift, the full sequence $(f^N)$ is
tight in the topology of uniform convergence over compact sets, and moreover
any weak subsequential limit possesses the Brownian Gibbs property. If in
addition the top curves converge in finite-dimensional distributions to the
parabolic $\mathrm{Airy}_2$ process, then a result of arXiv:2002.00684 implies
that $(f^N)$ converges to the parabolically shifted Airy line ensemble. These
results apply to a broad class of discrete jump distributions, including
geometric as well as any log-concave distribution whose support forms a compact
integer interval.
