There is crosslinguistic evidence that verbs of ‘coming’ and ‘going’ are prone to grammaticalize into sequential markers and, more generally, into textual connectives. This paper shows how, in the grammaticalization cline from directional deictics to sequentials, some of these markers have preserved their original semantics to varying degrees, while others have developed further into markers of past tense. The grammaticalization of ‘come’ and ‘go’ into markers of textual connectivity also exhibits a wide range of morphosyntactic variation: while in some languages they have retained all their formal verbal characteristics, in others they have become affixes, have followed the auxiliation or the serialization routes, or have fossilized. Three potential triggers for grammaticalization are discussed in the paper: the role of deicticity, the iteration hypothesis and the futurity scenario.