  We study the asymptotics of the $p$-mapping model of random mappings on $[n]$
as $n$ gets large, under a large class of asymptotic regimes for the underlying
distribution $p$. We encode these random mappings in random walks which are
shown to converge to a functional of the exploration process of inhomogeneous
random trees, this exploration process being derived (Aldous-Miermont-Pitman
2003) from a bridge with exchangeable increments. Our setting generalizes
previous results by allowing a finite number of ``attracting points'' to
emerge.
