  In this paper we prove a general form of the Mass Transference Principle for
$\limsup$ sets defined via neighbourhoods of sets satisfying a certain local
scaling property. Such sets include self-similar sets satisfying the open set
condition and smooth compact manifolds embedded in $\mathbb{R}^n$. Our main
result is applicable in locally compact metric spaces and allows one to
transfer Hausdorff $g$-measure statements to Hausdorff $f$-measure statements.
This work extends previous results of this type in several distinct directions.
