  We study families of varieties endowed with polarized canonical eigensystems
of several maps, inducing canonical heights on the dominating variety as well
as on the "good" fibers of the family. We show explicitely the dependence on
the parameter for global and local canonical heights defined by Kawaguchi when
the fibers change, extending previous works of J. Silverman and others.
Finally, fixing an absolute value $v \in K$ and a variety $V/K$, we descript
the Kawaguchi`s canonical local height $\hat{\lambda}_{V,E,\mathcal{Q},}(.,v)$
as an intersection number, provided that the polarized system $(V,\mathcal{Q})$
has a certain weak N\'{e}ron model over Spec$(\mathcal{O}_v)$ to be defined and
under some conditions depending on the special fiber. With this we extend
N\'{e}ron's work strengthening Silverman's results, which were for systems
having only one map.
