Exegesis:

kai ēspasato tēn Elisabet ‘and she greeted Elizabeth.’

aspazomai (also 10.4) ‘to greet,’ here of the person who enters, as the substantive aspasmos in 1.29, 41.

Translation:

And. Because the story leaves untold Mary’s arrival in the city it may be necessary to make the transition less abrupt, e.g. by preparing it in v. 39, cf. .’.. she arrived at a city in Judah, (40) and then she entered’ (Navajo); or by starting v. 40 with, ‘there,’ or, ‘on her arrival there.’

She entered and greeted, or, ‘having entered she greeted,’ ‘on-entering she greeted’ (Balinese). For greeted cf. on v. 29 and on “hail” in v. 28; Tboli has here, ‘called to,’ as one does when coming up the stairs of a house but not seeing anyone right away; Navajo has, ‘said peace/friendship to,’ a common way of reporting a friendly greeting, although ‘peace/friendship’ is not part of the actual formula of greeting.

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.