  Inspired by Gromov's partial differential relations, we introduce a notion of
differential transmutation, which allows to transfer some local properties of
solutions of a PDE to solutions of another PDE, in particular local
solvability, hypoellipticity, weak and strong unique continuation properties
and the Runge property. The latest refers to the possibility to approximate
some given local solutions by a global solution, with point force controls in
preassigned positions in the holes of the space domain. As examples we prove
that 2D Lam\'e-Navier system and the $3$D steady Stokes system, can be obtained
as differential transmutations of appropriate tensorizations of the Laplace
operator.
