Given a measure ${{\bar{\mu }}_{\infty }}$ on a locally symmetric space $Y=\Gamma \backslash G/K$ obtained as a weak-$*$ limit of probability measures associated with eigenfunctions of the ring of invariant differential operators, we construct a measure ${{\bar{\mu }}_{\infty }}$ on the homogeneous space $X=\Gamma \backslash G$ that lifts ${{\bar{\mu }}_{\infty }}$ and is invariant by a connected subgroup ${{A}_{1}}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then ${{\bar{\mu }}_{\infty }}$ is also the limit of measures associated with Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.