  We consider invasion percolation on the randomly-weighted complete graph
$K_n$, started from some number $k(n)$ of distinct source vertices. The outcome
of the process is a forest consisting of $k(n)$ trees, each containing exactly
one source. Let $M_n$ be the size of the largest tree in this forest. Logan,
Molloy and Pralat (arXiv:1806.10975) proved that if $k(n)/n^{1/3} \to 0$ then
$M_n/n \to 1$ in probability. In this paper we prove a complementary result: if
$k(n)/n^{1/3} \to \infty$ then $M_n/n \to 0$ in probability. This establishes
the existence of a phase transition in the structure of the invasion
percolation forest around $k(n) \asymp n^{1/3}$.
  Our arguments rely on the connection between invasion percolation and
critical percolation, and on a coupling between multi-source invasion
percolation with differently-sized source sets. A substantial part of the proof
is devoted to showing that, with high probability, a certain fragmentation
process on large random binary trees leaves no components of macroscopic size.
