The persistent random walk (PRW) offers an attractive means to describe suspension transport since its long-time behavior parallels classical diffusion (CD) and it devolves to the simple random walk (SRW) when there are equal probabilities to continue or change the direction of motion. Taylor dispersion and chromatography are described by PRW models. The PRW and its counterpart, the telegrapher's equation, have a finite propagation rate, i.e., a signal velocity, and hence differ from SRW and CD. In the fully developed region of blood flow, the motion of red blood cells (RBC) can be described by a PRW model with a spatially variable signal velocity. Estimates of the signal velocity are based on the general properties (symmetry) and shape of the hematocrit profile. The PRW model predicts the experimental observation that the effective viscosity in Fahraeus-Lindqvist experiments depends on the tubular hematocrit. The model indicates that flow-induced transport changes character slightly across the stream and is diffusive at long times; it expects significant flux (dispersion) at all lateral (radial) locations. If it is assumed that RBC motions are coupled to packet-like plasma motions and that material in the packets (e.g. latex beads) does not diffuse out as the RBCs and packets are shuffled, the new model is consistent with studies showing near-wall excesses of latex beads. The prior model of these events used a drift-diffusion (Smoluchowski) version of a random walk.