Alzheimer’s disease is characterized by accumulation of amyloid-β (Aβ) in the brain and in the walls of cerebral arteries. The focus of this work is on clearance of Aβ along artery walls, the failure of which may explain the accumulation of Aβ in Alzheimer’s disease. Periarterial basement membranes form continuous channels from cerebral capillaries to major arteries on the surface of the brain. Arterial pressure pulses drive peristaltic flow in the basement membranes in the same direction as blood flow. Here we forward the hypothesis that flexible structures within the basement membrane, if oriented such they present greater resistance to forward than retrograde flow, may cause net reverse flow, advecting Aβ along with it. A solution was obtained for peristaltic flow with low Reynolds number, long wavelength compared to channel height and small channel height compared to vessel radius in a Darcy–Brinkman medium representing a square array of cylinders. Results show that retrograde flow is promoted by high cylinder volume fraction and low peristaltic amplitude. A decrease in cylinder concentration and/or an increase in amplitude, both of which may occur during ageing, can reduce retrograde flow or even cause a transition from retrograde to forward flow. Such changes may explain the accumulation of Aβ in the brain and in artery walls in Alzheimer’s disease.