  Current gravitational-wave observations set the most stringent bounds on the
abundance of primordial black holes (PBHs) in the solar mass range. This
constraint, however, inherently relies on the merger rate predicted by PBH
models. Previous analyses have focused mainly on two binary formation
mechanisms: early Universe assembly out of decoupling from the Hubble expansion
and dynamical capture in present-day dark matter structures. Using reaction
rates of three-body processes studied in the astrophysical context, we show
that, under conservative assumptions, three-body interactions in PBH halos
efficiently produce binaries. Those binaries form at high redshift in
Poisson-induced PBH small-scale structures and a fraction is predicted to
coalesce and merge within the current age of the Universe, at odds with the
dynamical capture scenario where they merge promptly. In general, we find that
this channel predicts rates comparable to the dynamical capture scenario.
However, binaries formed from three-body interactions do not significantly
contribute to the overall PBH merger rate unless PBHs made up a dominant
fraction of the dark matter above the solar mass range, a scenario that is
ruled out by current constraints. Our results support strong bounds on the PBH
abundance in the stellar mass range derived from Laser Interferometer
Gravitational-Wave Observatory/Virgo/KAGRA observations. Finally, we show that
both dynamical channels are always subdominant compared to early Universe
assembly for PBH mergers in the asteroid mass range, while we expect it to
become relevant in scenarios where PBHs are initially strongly clustered.
