Palaeomagnetic evidence shows that the behaviour of the geodynamo has changed during geological times. These changes are visible through variations in the strength and stability of the magnetic dipole. Variations in the heat flux at the core-mantle boundary (CMB) due to mantle convection have been suggested as one possible mechanism capable of driving such a change of behaviour. This work aims at acquiring a more complete understanding of how lateral heterogeneities of the CMB heat flux affect the geodynamo while other relevant parameters are pushed towards realistic values. For this purpose, we ran geodynamo simulations with degree 1 and 2 spherical harmonic patterns of heat flux at the CMB. Several geodynamo models are used, ranging from standard numerical dynamos to more extreme parameters, including strong field cases and turbulent cases. We show that heat flux heterogeneities with amplitudes compatible with our knowledge of mantle convection history can favour multipolar dynamos. The multipolar transition is associated with a disruption of westward flows either through eastward thermal winds or through a loss of equatorial symmetry. Strong field dynamo models are found to have larger westward flows and are less sensitive to heat flux heterogeneities. Furthermore, we find that the dipolar fraction of the magnetic field correlates with $M_{Za}^*=\dfrac{\Lambda_{Za}}{Rm_{Za}^2}$ where $\Lambda_{Za}$ is the zonal antisymmetric Elsasser number and $Rm_{Za}$ is the zonal antisymmetric magnetic Reynolds number. Importantly, $M_{Za}^*$ estimated for the Earth's core is consistent with a reversing dipolar magnetic field. Within the range of $M_{Za}^*$ susceptible to reversals, breaking the equatorial symmetry or forcing eastward zonal flows through an equatorial cooling of the core consistently triggers reversals or a transition towards multipolar dynamos in our simulations. 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.