Dispersal heterogeneity is increasingly being observed in ecological populations and has long been suspected as an explanation for observations of non-Gaussian dispersal. Recent empirical and theoretical studies have begun to confirm this. Using an integro-difference model, we allow an individual's diffusivity to be drawn from a trait distribution and derive a general relationship between the dispersal kernel's moments and those of the underlying heterogeneous trait distribution. We show that dispersal heterogeneity causes dispersal kernels to appear leptokurtic, increases the population's spread rate, and lowers the critical reproductive rate required for persistence in the face of advection. Wavespeed has been shown previously to be determined largely by the form of the dispersal kernel tail. We qualify this by showing that when reproduction is low, the precise shape of the tail is less important than the first few dispersal moments such as variance and kurtosis. If the reproductive rate is large, a dispersal kernel's asymptotic tail has a greater influence over wavespeed, implying that estimating the prevalence of traits which correlate with long-range dispersal is critical. The presence of multiple dispersal behaviors has previously been characterized in terms of long-range versus short-range dispersal, and it has been found that rare long-range dispersal essentially determines wavespeed. We discuss this finding and place it within a general context of dispersal heterogeneity showing that the dispersal behavior with the highest average dispersal distance does not always determine wavespeed.