We examine the impact of surface inflows into activity belts on the operation of solar cycle models based on the Babcock-Leighton mechanism of poloidal field regeneration. Towards this end we introduce in the solar cycle model of Lemerle & Charbonneau (2017. ApJ 834: 133) a magnetic flux-dependent variation of the surface meridional flow based on the axisymmetric inflow parameterization developped by Jiang et al. (2010. ApJ 717: 597). The inflow dependence on emerging magnetic flux thus introduces a bona fide nonlinear backreaction mechanism in the dynamo loop. For solar-like inflow speeds, our simulation results indicate a decrease of 10-20% in the strength of the global dipole building up at the end of an activity cycle, in agreement with earlier simulations based on linear surface flux transport models. Our simulations also indicate a significant stabilizing effect on cycle characteristics, in that individual cycle amplitudes in simulations including inflows show less scatter about their mean than in the absence of inflows. Our simulations also demonstrate an enhancement of cross-hemispheric coupling, leading to a significant decrease in hemispheric cycle amplitude asymmetries and temporal lag in hemispheric cycle onset. Analysis of temporally extended simulations also indicate that the presence of inflows increases the probability of cycle shutdown following an unfavorable sequence of emergence events. This results ultimately from the lower threshold nonlinearity built into our solar cycle model, and presumably operating in the sun as well.