  Van der Waals heterostructures provide a versatile tool to not only protect
or control, but also enhance the properties of a 2D material. We use ab initio
calculations and semi-analytical models to find strategies which boost the
mobility of a current-carrying 2D semiconductor within an heterostructure.
Free-carrier screening from a metallic "screener" layer remotely suppresses
electron-phonon interactions in the current-carrying layer. This concept is
most effective in 2D semiconductors whose scattering is dominated by screenable
electron-phonon interactions, and in particular the Fr\"ohlich coupling to
polar-optical phonons. Such materials are common and characterised by overall
low mobilities in the small doping limit, and much higher ones when the 2D
material is doped enough for electron-phonon interactions to be screened by its
own free carriers. We use GaSe as a prototype and place it in a heterostructure
with doped graphene as the "screener" layer and BN as a separator. We develop
an approach to determine the electrostatic response of any heterostructure by
combining the responses of the individual layers computed within
density-functional perturbation theory. Remote screening from graphene can
suppress the long-wavelength Fr\"ohlich interaction, leading to a consistently
high mobility around $500$ to $600$ cm$^2$/Vs for carrier densities in GaSe
from $10^{11}$ to $10^{13}$ cm$^{-2}$. Notably, the low-doping mobility is
enhanced by a factor 2.5. This remote free-carrier screening is more efficient
than more conventional manipulation of the dielectric environment, and it is
most effective when the separator (BN) is thin.
