  In the present article, we define a notion of good height functions on
quasi-projective varieties $V$ defined over number fields and prove an
equidistribution theorem of small points for such height functions. Those good
height functions are defined as limits of height functions associated with
semi-positive adelic metrization on big and nef $\mathbb{Q}$-line bundles on
projective models of $V$ satisfying mild assumptions.
  Building on a recent work of the author and Vigny as well as on a classical
estimate of Call and Silverman, and inspiring from recent works of K\"uhne and
Yuan and Zhang, we deduce the equidistribution of generic sequence of
preperiodic parameters for families of polarized endomorphisms with marked
points.
