  Alongside the slow (s) and rapid (r) neutron capture processes, an
intermediate neutron capture process (i-process) is thought to exist. It
happens when protons are mixed in a convective helium-burning zone, and is
referred to as proton ingestion event (PIE). A possible astrophysical site is
the asymptotic giant branch (AGB) phase of low-mass low-metallicity stars. We
provide i-process yields of a grid of AGB stars experiencing PIEs. We computed
12 models with initial masses of 1, 2, and 3 $M_{\odot}$ and metallicities of
[Fe/H] $=-3.0$, $-2.5$ $-2.3,$ and $-2.0, $ with the stellar evolution code
STAREVOL. We used a nuclear network of 1160 species at maximum, coupled to the
chemical transport equations. These simulations do not include any extra mixing
process. Proton ingestion takes place in six out of our 12 AGB models. These
models experience i-process nucleosynthesis characterized by neutron densities
of $\simeq 10^{14} -10^{15}$ cm$^{-3}$. Depending on the PIE properties two
different evolution paths follow: either the stellar envelope is quickly lost
and no more thermal pulses develop or the AGB phase resumes with additional
thermal pulses. This behaviour critically depends on the pulse number when the
PIE occurs, the mass of the ingested protons, and the extent to which the pulse
material is diluted in the convective envelope. The surface enrichment after a
PIE is a robust feature of our models and it persists under various convective
assumptions. Our models can synthesise heavy elements up to Pb without any
parametrized extra mixing process such as overshoot or inclusion of a
$^{13}$C-pocket. Nevertheless, it remains to be explored how the i-process
depends on mixing processes, such as overshoot, thermohaline, or rotation.
