  The goal of the paper is to introduce a new set of tools for the study of
discrete and continuous $\beta$-corners processes. In the continuous setting,
our work provides a multi-level extension of the loop equations (also called
Schwinger-Dyson equations) for $\beta$-log gases obtained by Borot and Guionnet
in (Commun. Math. Phys. 317, 447-483, 2013). In the discrete setting, our work
provides a multi-level extension of the loop equations (also called Nekrasov
equations) for discrete $\beta$-ensembles obtained by Borodin, Gorin and
Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).
