We develop a black-box theory, which can be used to show that a sequence of Gibbsian line ensembles is tight, provided that the one-point marginals of the lowest labeled curves of the ensembles are tight and globally approximate an inverted parabola. Our theory is applicable under certain technical assumptions on the nature of the Gibbs property and the underlying random walk measure. As a particular application of our general framework we show that a certain sequence of Gibbsian line ensembles, which naturally arises in the log-gamma polymer, is tight in the ubiquitous KPZ class $1/3: 2/3$ scaling, and also that all subsequential limits satisfy the Brownian Gibbs property, introduced by Corwin and Hammond in (Invent. Math. 195, 441-508, 2014). One of the core results proved in the paper, which underlies many of our arguments, is the construction of a continuous grand monotone coupling of Gibbsian line ensembles with respect to their boundary data (entrance and exit values, and bounding curves). Continuous means that the Gibbsian line ensemble measure varies continuously as one varies the boundary data, grand means that all uncountably many measures (one for each boundary data) are coupled to the same probability space, and monotone means that raising the values of the boundary data likewise raises the associated measure. Our continuous grand monotone coupling generalizes an analogous construction, which was recently implemented by Barraquand, Corwin and Dimitrov in (arXiv:2101.03045), from line ensembles with a single curve to ones with an arbitrary finite number of curves.