A summary of the "Instruction de l’épiscopat catholique de Grèce sur l’adoption du Symbole dit de Nicée-Constantinople dans la liturgie latine en langue grecque" as published in French in Istina 28, no. 3 (1983): 319-325. Note that this seems to have been a ruling for Catholics of the Latin rite, not just Catholics of the Byzantine rite. From pp. 321-322:
the crux of the problem . . . resides in the fact that the verbs procedere and ἐκπορεύειν are not exactly synonymous. Procedere translates, in fact, other Greek verbs such as προέρχομαι and ἐξέρχομαι (see, for example, the Latin texts of the Vetus latina or of the Vulgate for Jn 8:42, where ἐξέρχομαι is translated by procedere). One finds the inverse verification of this affirmation in the fact that, since Tertullian (d. 245), the Latin tradition employs the verb procedere (προέρχομαι) generically in order to express as much the Son’s as the Spirit’s relation of origin, and, thus, speaks of [the] processio (προέλευσις) of the Son and of [the] processio (προέλευσις) of the Holy Spirit, while the Greek tradition employs ἐκπορεύειν solely for designating specifically the relation of the Holy Spirit to the Paternal principle in as much as th[at relation] is distinguished from [the relation involved in] the generation of the Son. This different usage of the two verbs in the two traditions is explained by the fact that the two verbs in their respective languages have different semantic nuances. The fact of the matter is that the Greek ἐκπορεύειν—which signifies exactly 'to go out from the door' or 'from the first source'—references rather the origin from whence the thing which proceeds comes, while the Latin procedere—which wishes to say precisely 'to proceed'—has in view rather the very thing which proceeds without considering whether the source from which it proceeds is an ultimate [(première)] source or not. . . . [Thus] language has played an important role in the formation and structuring of the two patristic traditions, oriental and occidental. The Latin fathers can utilize the verb procedere in a generic fashion to designate the two Persons who come from the Father, ultimate source of the Trinity. And yet, in order to safeguard the Monarchy of the Father in the Trinitarian relations, the Latin tradition, even while affirming that the Holy Spirit proceeds (προέρχεται or πρόεισι) from the Father and the Son, has always underscored, in the wake of St. Augustine, that the Spirit proceeds from the Father principaliter (πρωταρχικῶς). The Greek fathers, whose verb ἐκπορεύομαι implies always the idea of a ‘going out from the ultimate source’, have, on their side, never said that the Holy Spirit ἐκπροεύεται καὶ τοῦ Υἱοῦ. They could not say this, for that formula would have signified that they considered the Son as being, he, too, an ultimate source of the Trinity, which would constitute a dogmatic error. And yet, the doctrine according to which the Holy Spirit comes forth [(provient)] also from the Son (καὶ ἐκ τοῦ Υἱοῦ) is found clearly expressed in the tradition of the Greek fathers, but always with other formulas. The school of Alexandria, with St. Cyril at its head, affirms that the Holy Spirit πρόεισι ἐκ τοῦ Πατρὸς καὶ Υἱοϋ. The Cappadocians and St. John of Damascus prefer to preserve the original verb that one finds in the Gospels, ἐκπορεύομαι, and, in order to express the same truth, to employ the phrase διὰ τοῦ Υἱοῦ: τὸ Πνεῦμα τὸ Ἅγιον ἐκ τοῦ Πατρὸς διὰ τοῦ Υἱοῦ ἐκπορευόμενον.
Each of these two formulas, Spiritus Sanctus qui ex Patre Filioque procedit and to τὸ Πνεῦμα τὸ Ἅγιον ἐκ τοῦ Πατρὸς διὰ τοῦ Υἱοῦ ἐκπορευόμενον, in its own language and in its own theological system, express in [an] identical fashion the same faith.