Direct numerical simulations of fully-developed turbulent flow through a straight square duct with increasing corner rounding radius r were performed to study the influence of corner geometry on the secondary flow. Unexpectedly, the increased rounding of the corners from r = 0 to 0.75 does not lead to a monotonic trend towards the pipe case of r = 1. Instead, the secondary vortices relocate close to the region of wall-curvature change. This behavior is connected to the inhomogeneous interaction between near-wall bursting events, which are further characterized in this work with the definition of their local preferential direction. We compare our results with those obtained for the flow through a square duct (which corresponds to r = 0) and through a round pipe (r = 1), focusing on the influence of r on the wall-shear stress distribution and the turbulence statistics along the centerplane and the corner bisector. The former shows that high-speed streaks are preferentially located near the transition between straight and curved surfaces. The Reynolds numbers based on the centerplane friction velocity and duct half-height are Re-r,Re-