  Let $S$ be a closed oriented surface of genus at least $2$. Using the
parameterisation of the deformation space of globally hyperbolic maximal
anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle over
the Teichm\"uller space of $S$, we study the behaviour of these geometric
structures along pinching sequences. We show, in particular, that the regular
globally hyperbolic anti-de Sitter structures introduced in arXiv:1806.08176
naturally appear as limiting points.
