The process in which nicotinamide mononucleotide is transported from one side of a membrane to the other by means of some agent such as a transporter or pore. Nicotinamide mononucleotide is a ribonucleotide in which the nitrogenous base, nicotinamide, is in beta-n-glycosidic linkage with the c-1 position of d-ribose. It is a constituent of NAD and NADP.