  In this paper, several convergence results for fine $p$-(super)minimizers on
quasiopen sets in metric spaces are obtained. For this purpose, we deduce a
Caccioppoli-type inequality and local-to-global principles for fine
$p$-(super)minimizers on quasiopen sets. A substantial part of these
considerations is to show that the functions belong to a suitable local fine
Sobolev space. We prove our results for a complete metric space equipped with a
doubling measure supporting a $p$-Poincar\'e inequality with $1<p< \infty$.
However, most of the results are new also for unweighted $\mathbf{R}^n$.
