  Fix $p\geq 5$ an odd integer integer. Let $M_n$ be a uniform $p$-angulation
with $n$ vertices and endowed with the uniform probability measure on its
vertices. We prove that, there exists $C_p\in \mathbb{R}_+$ such that, after
rescaling distances by $C_p/n^{1/4}$, $M_n$ converges in distribution for the
Gromov-Hausdorff-Prokhorov topology towards the Brownian map. To prove the
preceding fact, we introduce a `bootstrapping' principle for distributional
convergence of random labelled plane trees. In particular, the latter allows to
obtain an invariance principle for labeled multitype Galton-Watson trees, with
only a weak assumption on the centering of label displacements
