History: the griffin-demon does not stem from Babylonia; there he is attested first on the assyrianizing robe of Nabû-mukîn-apli (cf. Brinkman PHPKB 171, beginning of the 1st millennium) holding cone and bucket (King BBSt Pl. LXXIV); in Assyria, Syria, and the north he is attested much earlier (Parker Essays Wilkinson 33, Collon AOAT 27 222, on MAss seals: Klengel Brandt FuB 10 2438,39), cf. Madhloom Sumer 20 57ff.
Thus we are led to believe that a traditional northern hybrid with apotropaic functions was matched in Assyria with a traditional Babylonian literary figure with similar functions.
In Babylonia, from MB onwards, the apotropaic apkallū were viewed as partly man and partly carp; in the early first millennium Babylonia takes over the bird-apkallū (BBSt Pl. LXXIV), and Assyria the fish-apkallū (Rittig Kleinplastik 87).
The first millennium magical texts of Babylonian origin had to accommodate these foreign apotropaic beings.
The bird-apkallū are accommodated in bit mēseri by a slight change of form and sequence of the names of the fish-apkallū (text III.B.10). In ritual I/II they are simply provided with the same incantation as the fish-apkallū.
F.A.M. Wiggermann, Mesopotamian Protective Spirits: The Ritual Texts, STYX&PP Publications, Groningen, 1992, p. 75-6.