In infinite dimensions, many-body systems of pairwise interacting particles provide exact analytical benchmarks for the features of amorphous materials, such as the stress--strain curve of glasses under quasistatic shear. Here, instead of global shear, we consider an alternative driving protocol, as recently introduced by Morse et al 2020 (arXiv:2009.07706), which consists of randomly assigning a constant local displacement on each particle, with a finite spatial correlation length. We show that, in the infinite-dimensional limit, the mean-field dynamics under such a random forcing are strictly equivalent to those under global shear, upon a simple rescaling of the accumulated strain. Moreover, the scaling factor is essentially given by the variance of the relative local displacements of interacting pairs of particles, which encodes the presence of a finite spatial correlation. In this framework, global shear is simply a special case of a much broader family of local forcing, which can be explored by tuning its spatial correlations. We discuss the specific implications for the quasistatic driving of glasses---initially prepared at a replica-symmetric equilibrium---and how the corresponding `stress--strain'-like curves and elastic moduli can be rescaled onto their quasistatic-shear counterparts. These results hint at a unifying framework for establishing rigorous analogies, at the mean-field level, between different driven disordered systems.