Self-avoiding walks in 2-d and 3-d lattices are implemented as models of polymer propagation in confined media like vesicle bilayers and micelles. Simulations conducted on such microreactors of various sizes (total number of monomers N(tot) between 80 and 10(6)) and shapes show that the propagation rate essentially depends on the ratio of the degree of polymerization (D.P.) and N(tot). A scaling function is proposed giving excellent agreement with the simulations : it features a decrease of the rate of propagation arising from a direct boundary effect, related to the contact between the radical and the impermeable boundaries of the lattice, and from an exponential factor, accounting for the confinement effect. Taking into account this rate dependence on the D.P., length and mass distributions are shown to be increasing functions of the D.P. when the rate of termination (by transfer) is small enough. In this limit, these distributions present a sharp increase in the vicinity of a D.P. about N(tot). This accumulation of polymers at a D.P. close to N(tot) entails a significant decrease of the polydispersity.