We consider the one-dimensional initial value problem for the viscous
transport equation with nonlocal velocity utβ=uxxββ(u(Kβ²βu))xβ with a given kernel Kβ²βL1(R). We show the existence
of global-in-time nonnegative solutions and we study their large time
asymptotics. Depending on Kβ², we obtain either linear diffusion waves ({\it
i.e.}~the fundamental solution of the heat equation) or nonlinear diffusion
waves (the fundamental solution of the viscous Burgers equation) in asymptotic
expansions of solutions as tββ. Moreover, for certain aggregation
kernels, we show a concentration of solution on an initial time interval, which
resemble a phenomenon of the spike creation, typical in chemotaxis models