Rising and falling diphthongs in dialects
Diphthongs, both rising and falling, are widespread in the dialects. At the same time levelling of diphthongs also occurs. There are four possibilities:

a)  a "semivowel" is lost;
b)  a "semivowel" turns into a consonant;
c) two strong (full) vowels are represented in the position of a diphthong;
d) A diphthong may merge into one sound, resulting in the formation of a single vowel

(Besarion Jorbenadze, The Kartvelian Languages and Dialects, Tbilisi, Mecniereba, 1991, pp. 48-49).

Rising and falling diphthongs in Old Georgian
Falling diphthongs used to be regular on the boundary of morphemes:
a)  in the declension system of nouns ending in a vowel in Old Georgian, usually represented with the sequence of a, e, o, u vowels and j "semivowel".  The disappearance of the diphthongs in the declension influenced the case markers (c'q'aro-jsa - c'q'aro-sa - c'q'aro-si).
b) in the superlative degree of adjectives in Old Georgian, usually in the ej sequence, as in umdzhobe-js-i, and in some oj stems, as in upro-js.
c) in some adverbs: kwemo-jt, c'ina-jt (the instrumental case by origin).

Falling diphthongs within a root are attested in a few cases, only in borrowed stems: Israejl-, Ierusalejm-,  amejn, shejn- (build), cejn- (spring up).

Rising diphthongs in Old Georgian are found only within a morpheme and consist of the semivowel w, on the one hand, and the vowels a, e, i, on the other  hand: twali, ttwe, shwidi.
(Ak'ak'i Shanidze, Dzveli kartuli ena, The Old Georgian language, Tbilisi, 1978, Zurab Sardzhveladze, Dzveli kartuli ena, The Old Georgian language, Tbilisi, 1997)