  For non-negative integers $(d_n(k))_{k \ge 1}$ such that $\sum_{k \ge 1}
d_n(k) = n$, we sample a bipartite planar map with $n$ faces uniformly at
random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge
1$ and we study its asymptotic behaviour as $n \to \infty$. We prove that the
diameter of such maps grow like $\sigma_n^{1/2}$, where $\sigma_n^2 = \sum_{k
\ge 1} k (k-1) d_n(k)$ is a global variance term. More precisely, we prove that
the vertex-set of these maps equipped with the graph distance divided by
$\sigma_n^{1/2}$ and the uniform probability measure always admits
subsequential limits in the Gromov-Hausdorff-Prokhorov topology.
  Our proof relies on a bijection with random labelled trees; we are able to
prove that the label process is always tight when suitably rescaled, even if
the underlying tree is not tight for the Gromov-Hausdorff topology. We also
rely on a new spinal decomposition which is of independent interest. Finally
this paper also serves as a toolbox for a companion paper in which we discuss
more precisely Brownian limits of such maps.
