Tl;dr: If any, such claims probably came from the 'propaganda' of the archbishop of Canterbury that sought to establish his jurisdiction all over the British Isles (not only southern England). The Irish church mainly independently got more in line with the Catholic church in other Latin West at least since ca. 1100 (Cf. Holland 2000)
While recent researches have still hotly debated its authenticity and the background of the issue of the document itself, the majority of the relevant literature agree that this alleged papal bull (Laudabiliter) is indeed a difficult one and should not be taken at face value (Flanagan 2010: 5; Duggan 2003: 138-52).
The whole text of Laudabiliter is only extant in Conquest of Ireland (Expugnatio Hibernica), written by Gerald of Wales in 1189. He cites this papal bull with another alleged papal bull, Quoniam ea, issued by Pope Alexander III also to authorize Henry's conquest in Ireland, but at least the latter is a 'crude fake' (Duggan 2003: 140). Therefore, we should always take the possibility in mind that the extant text of the former papal bull, Laudabiliter, was probably heavily altered to reflect Gerald's agenda if the original papal bull had really been issued (it was not impossible) (Duggan 2003: 148f.).
Regardless of the authenticity of the bull itself, the most popular hypothesis behind the bull Laudabiliter is that the issue of this document was a joint plot by the Pope Adrian IV (the English) and the English church, especially archbishop of Canterbury, to lure young King Henry of England into their expansion plan to integrate Ireland into the Anglo-Norman orbit both politically and ecclesiastically (Downham 2018: 281). The Pope could also expect the tax revenue as well as the more strict obedience from the Irish Church (So, a little amount of the payment of Peter's Pence could be listed as an example of such 'drifts' mentioned in OP, I suppose).
After the Norman conquest, archbishops of Canterbury tried to advance his formal jurisdiction to the primate of all the other bishops in the British Isles. The first Norman archbishop (though himself was an Italian) of Canterbury, Lanfranc, to give an example, sought the oath of obedience from the bishop of Dublin while consecrating him by himself. He and other Norman clergy brought many Continental monastic texts as well as continental-style Canon law collections into the conquered England so that they also wish to introduce more hierarchical, latin-text based ecclesiastical organization, in contrast to more vernacular-based and regionally distinct Anglo-Saxon as well as traditional British churches.
The 'church reform' could indeed have been a convenient banner for them to introduce the more uniform rules under the auspices of archbishop of Canterbury, regardless of the extent of the actual divergences found in the local ecclesiastical practices. Even the more hierarchical church order, headed by the archbishop and its possible superior (primate) was sometimes regarded as a purpose of such a reform (Cf. Brett 2006: 21f.).
While the concept of 'church reform' itself was gradually accepted across the British Isles, represented by new trend within the church, such as the new monasticism like the Cistercian order Augustinian Canons, however, attached claims of the archbishop of Canterbury (as a primate of all the British church) met vehement resistance from the local churchmen here and there. First, archbishop of York opposed them (and won his own claim over Canterbury in 1120s), then, the Irish succeeded in inviting Papal Legate John Paparo, dispatched by Pope Eugene III into Ireland, by the arbitration of Malachy the Irish, a friend of the very influential churchman at that period, Bernard of Clairvaux [of the Cistercians]. Even the Pope Eugene III himself had been one of Bernard's disciples.
Ireland had already had two archbishoprics of Armagh and Cashel since 1111, but the reformed synod at Kells-Mellifont in 1152 presided by Legate John Paparo also promoted two further bishops, namely Dublin and Tuam, to the archbishops, though under the nominal auspices of the primate of Armagh. This synod's action, totally sanctioned by the papal legate from Pope Eugene III, would have been a death sentence to the political-jurisdictional ambition of the archbishop of Canterbury, so the English bishops must have harried to take a counter-action against this measure in the middle of the 1150s. It was the alleged bull Laudabiliter, as the most popular hypothesis suppose.
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