Unsteady heat or mass transport from a particle with an arbitrary shape suspended in a fluid of infinite expanse is considered in the limit of small Péclet numbers where diffusion is dominant. In a frame of reference in which the particle appears to be stationary, the velocity of the fluid is uniform or varies in a linear manner with respect to the spatial coordinates, with an arbitrary time dependence. The temperature or concentration of a species at the surface of the particle is held at a certain constant value, whereas that at infinity is held at another constant value. Two particular problems are considered, both to leading order with respect to the Péclet number: (a) the rate of transport from a particle that is introduced suddenly into a steady flow near the steady state; and (b) the average rate of transport from a particle that is suspended in a time-periodic flow. The theory uses the method of matched asymptotic expansions and employs the Green's function of the convection–diffusion equation for a generally unsteady uniform or linear flow. The Green's function is derived in closed form by first performing a transformation to a Lagrangian framework. In the first problem of transient transport, it is found that the functional form of the rate of transport near the steady state is affected strongly by the structure of the incident flow: the decay in uniform or elongational flow is exponential, whereas the decay in simple shear flow is algebraic. In the second problem of transport in a periodic flow, it is found that the value of a properly defined frequency parameter has a strong influence on the mean rate of transport, for all types of flow. The oscillation induces convective mixing and thereby reduces the mean rate of transport by a substantial factor. The ability of the theory to describe another situation of heat or mass transport considered by Pedley is also discussed.