We analyse the properties of circumgalactic gas around simulated galaxies in the redshift range z ≥ 3, utilizing a new sample of cosmological zoom simulations. These simulations are intended to be representative of the observed samples of Lyman α (Ly α) emitters recently obtained with the multi unit spectroscopic explorer (MUSE) instrument (halo masses ∼10¹⁰-10¹¹ M_⊙). We show that supernova feedback has a significant impact on both the inflowing and outflowing circumgalactic medium (CGM) by driving outflows, reducing diffuse inflow rates, and by increasing the neutral fraction of inflowing gas. By temporally stacking simulation outputs, we find that significant net mass exchange occurs between inflowing and outflowing phases: none of the phases are mass-conserving. In particular, we find that the mass in neutral outflowing hydrogen declines exponentially with radius as gas flows outwards from the halo centre. This is likely caused by a combination of both fountain-like cycling processes and gradual photoionization/collisional ionization of outflowing gas. Our simulations do not predict the presence of fast-moving neutral outflows in the CGM. Neutral outflows instead move with modest radial velocities (∼50 km s^-1), and the majority of the kinetic energy is associated with tangential rather than radial motion.