Low literacy and schooling rates are a significant factor among adults in the recent migration towards Europe. Furthermore, migrants experience social marginalisation and spatial segregation in the new countries and, hence, they are low-exposed to the target language. Research on interlanguages development in such sociolinguistic contexts is still peripheral. Within the little existing research, there is consensus that adult learners with none/limited literacy acquire L2 linguistic competence more slowly compared with educated adults, but there are divided views on the relationship between L1(s) literacy and L2 acquisition (Tarone et al. 2009; Vainikka et al. 2017; Young-Scholten and Strom 2006). Slow acquisition, in fact, may result from limited literacy or from other factors deriving from literacy, e.g. low exposure to the target language and no access to written texts. Whether and how these sociolinguistic variables influence L2 acquisition (and, if so, what their respective impact is) still need to be verified. Against this background, a longitudinal study was carried out at the University of Palermo, Italy, in 2017-2019, involving 20 adult migrants from sub-Saharan Africa and Bangladesh. Half of them had little/no literacy prior to arrival. The data collected over 13 months, through interviews and narrative tasks, were analysed from a functionalist perspective inspired by the basic variety model (Klein and Perdue 1997; Giacalone Ramat 2003). Data analysis, focused on the acquisition of L2 tense, aspect and modality, brought to light a general low development of the participants’ interlanguages, which barely reach the early stages of the post-basic continuum. This results from the common scenario of very low exposure to the target language, while learners’ degree of literacy does not appear to play a role, as literate and low/non-literate learners follow the same path of morphosyntax acquisition. However, literacy acts in a subtler way, favouring the development of specific morphosyntactic sub-patterns. This involves the analytical phase that precedes the morphological encoding of verbal categories in the transition from basic to post-basic varieties. At this stage, grammatical and lexical information are encoded separately, possibly by means of non-target constructions in which they are distributed among the diverse constituents (Benazzo 2003; Benazzo and Starren 2007; Starren 2001). Some of these have already been described for L2 Italian, e.g. auxiliary constructions (non è:be.3SG credere:INF ‘he does not believe’, Bernini 2003). Others have escaped the attention of specialists, e.g. the light verb construction (fare:INF mangiare:INF ‘I eat’). Non-target constructions are temporary grammaticalisation strategies to cover the functional space of forms (bound morphemes) not yet acquired. Non-target constructions especially occur in learners with limited literacy, while literate learners use them in a more sporadic and transient way. This can be interpreted as an effect of reduced exposure to the input as a consequence of limited literacy. As learners with limited literacy are exposed exclusively to oral input, they struggle in identifying bound morphemes in the input, due to the low salience, redundancy and frequent reduction phenomena these formatives undergo in spontaneous speech. This leads them to favour and maintain “heavier” constructions, made of material more easily perceived in the input.