There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian fibrations, as described in Definition 4.1 of Asaf Horev and Lior Yanovski: On Conjugates and Adjoint Descent. An alternative construction is described in David Gepner, Rune Haugseng, and Thomas Nikolaus: Lax Colimits and Free Fibrations in ∞-Categories.

Similarly in the context of model categories, one can construct lax limits using sections, as described in Definition 2.21 of Clark Barwick: On left and right model categories and left and right bousfield localizations. For a similar account see Definition 3.1 in Julia E. Bergner : Homotopy limits of model categories and more general homotopy theories.

Are all these models equivalent? In particular, is there an equivalence that allows me to move between lax limit of a (small) diagram of model categories to the lax limit of the corresponding diagram of presentable infinity categories.

I suspect a similar assertion holds, based on remark 2.24 in the paper above by Clark Barwick. However, I have not been able to find a reference supporting this claim. Thanks in advance.