  Aims: We aim to investigate how converging flows towards active regions
affect the surface transport of magnetic flux, as well as their impact on the
generation of the Sun's poloidal field. The inflows constitute a potential
non-linear mechanism for the saturation of the global dynamo and may contribute
to the modulation of the solar cycle in the Babcock-Leighton framework.
  Methods: We build a surface flux transport code incorporating a parametrized
model of the inflows and run simulations spanning several cycles. We carry out
a parameter study to assess how the strength and extension of the inflows
affect the build-up of the global dipole field. We also perform simulations
with different levels of activity to investigate the potential role of the
inflows in the saturation of the global dynamo.
  Results: We find that the interaction of neighbouring active regions can lead
to the occasional formation of single-polarity magnetic flux clumps
inconsistent with observations. We propose the darkening caused by pores in
areas of high magnetic field strength as a plausible mechanism preventing this
flux-clumping. We find that inflows decrease the amplitude of the axial dipole
moment by a $\sim30\,\%$, relative to a no-inflows scenario. Stronger (weaker)
inflows lead to larger (smaller) reductions of the axial dipole moment. The
relative amplitude of the generated axial dipole is about $9\%$ larger after
very weak cycles than after very strong cycles. This supports the inflows as a
non-linear mechanism capable of saturating the global dynamo and contributing
to the modulation of the solar cycle within the Babcock-Leighton framework.
