  The intermediate neutron capture process (i-process) can develop during
proton ingestion events (PIE), potentially during the early stages of low-mass
low-metallicity asymptotic giant branch (AGB) stars. We examine the impact of
overshoot mixing on the triggering and development of i-process nucleosynthesis
in AGB stars of various initial masses and metallicities. We computed AGB
stellar models, with initial masses of 1, 2, 3, and 4 M$_{\odot}$ and
metallicities in the $-2.5 \le $ [Fe/H] $\le 0$ range, using the stellar
evolution code STAREVOL with a network of 1160 nuclei coupled to the transport
equations. We considered different overshooting profiles below and above the
thermal pulses, and below the convective envelope. The occurrence of PIEs is
found to be primarily governed by the amount of overshooting at the top of
pulse ($f_{\rm top}$) and to increase with rising $f_{\rm top}$. For $f_{\rm
top} =$ 0, 0.02, 0.04, and 0.1, we find that 0 %, 6 %, 24 %, and 86 % of our 21
AGB models with $-2<$ [Fe/H] $<0$ experience a PIE, respectively. We also find
that PIEs leave a $^{13}$C-pocket at the bottom of the pulse that can give rise
to an additional radiative s-process nucleosynthesis, and ultimately produce a
noticeable mixed i+s chemical signature at the surface. Finally, the chemical
abundance patterns of 22 observed r/s-stars candidates with $-2<$ [Fe/H] $<-1$
are found to be in reasonable agreement with our AGB model predictions. The
binary status of the dwarfs/giants being unclear, we suggest that these stars
have acquired their chemical pattern either from the mass transfer of a
now-extinct AGB companion or from an early generation AGB star that polluted
the natal cloud. Stricter constraints from multi-dimensional hydrodynamical
models on overshoot coefficients could deliver new insights into the
contribution of AGB stars to heavy elements in the Universe.
