The class of moving-average fractional Lévy motions (MAFLMs), which are fields parameterized by a d-dimensional space, is introduced. MAFLMs are defined by a moving-average fractional integration of order H of a random Lévy measure with finite moments. MAFLMs are centred d-dimensional motions with stationary increments, and have the same covariance function as fractional Brownian motions. They have H-d/2 Hölder-continuous sample paths. When the Lévy measure is the truncated random stable measure of index α, MAFLMs are locally self-similar with index \widetilde{H} =H -d/2+d/ α. This shows that in a non-Gaussian setting these indices (local self-similarity, variance of the increments, Hölder continuity) may be different. Moreover, we can establish a multiscale behaviour of some of these fields. All the indices of such MAFLMs are identified for the truncated random stable measure.