The article investigates diachronic changes in infinitival complementation from Caucasian Albanian to modern Udi dialects. It describes the syntactic structure of infinitival complements in Caucasian Albanian, 19th century Vartashen Udi and two modern dialects, and concentrates on case marking of overt subjects in constructions with the matrix verbs ‘can, be able’, ‘begin’ and ‘want’. From a diachronic point of view, the data presented in the article allow us to conclude that historical changes in both the lexical form of complement-taking predicates and the morphology of their complements obey Cristofaro’s (2003) Complement Deranking Hierarchy.