Context. Recent observations suggest a double-branch behaviour of Li/H versus metallicity in the local thick and thin discs. This is reminiscent of the corresponding O/Fe versus Fe/H behaviour, which has been explained as resulting from radial migration in the Milky Way disc. Aims. We study here the role of radial migration in shaping these observations. Methods. We use a semi-analytical model of disc evolution with updated chemical yields and parameterised radial migration. We explore the cases of long-lived (red giants of a few Gy lifetime) and shorter-lived (asymptotic giant branch stars of several 108 yr) stellar sources of Li, as well as those of low and high primordial Li. We show that both factors play a key role in the overall Li evolution.Results. We find that the observed two-branch Li behaviour is only directly obtained in the case of long-lived stellar Li sources and low primordial Li. In all other cases, the data imply systematic Li depletion in stellar envelopes, thus no simple picture of the Li evolution can be obtained. This concerns also the reported Li/H decrease at supersolar metallicities.