Palaeomagnetic evidence shows that the behaviour of the geodynamo has changed during geological times. Variations in the heat flux at the core-mantle boundary (CMB) due to mantle convection could be responsible. Previous studies, based on unrealistically viscous dynamo simulations, have shown that large-scale CMB heat flux heterogeneities impact the magnetic dipole stability. To better understand how they affect the geodynamo, we used several simulations, ranging from standard numerical dynamos to more extreme parameters, including strong-field cases and turbulent cases. We show that heterogeneities with realistic amplitudes can favour a multipolar dynamo by either forcing equatorially antisymmetric zonal flows or eastward zonal flows. Strong-field dynamo models are found to be less sensitive, due to significant westward flows. We also find that the dipolar fraction of the magnetic field is best captured by $M^*=M\ E_{\eta}\ \dfrac{l_c}{\pi}$ where $M$ is the magnetic to kinetic energy ratio, $E_{\eta}$ is the magnetic Ekman number, and $l_c$ is the dominant harmonic degree of the flow, with multipolar dynamos found at lower $M^*$. $M^*$ estimated for the Earth's core is consistent with a reversing dipolar magnetic field. Within the range of $M^*$ susceptible to reversals, breaking the equatorial symmetry or forcing eastward zonal flows in our simulations consistently triggers reversals or a transition towards multipolar dynamos. Our results support that time variations of heat-flux heterogeneities driven by mantle convection through Earth's history are capable of inducing the significant variations in the reversal frequency observed in the palaeomagnetic record.