It is not obvious that forms of the type (wV)d meaning “to, towards” are related to this preposition (Arakelova 2001:22). They are united into a separate lexeme, see *ˀVd/d.
Akkadian
adi - up to, as far as; until, up to (connoting time); concerning, as to
(CAD A₁ 115, AHw. 12)
Also adu, gadu, ḳаdu, ḫadu, ad. The background of variants with initial ḳ and g is difficult to establish.
GAG § 114 j analyzes adīni and adīšu as lexicalized combinations of adi and pronominal suffixes.
ˁad - as far as, until, just before, during
(HALOT 786)
ˁădē in open pretonic syllables, ˁāday in non-pretonic ones. The form ˁădē (< *ˁaday) attested in poetic texts corresponds directly to the protoform. Originally a plural noun, it is used in plural construct state when combined with pronominal suffixes (Gesenius 1910:304).