As far as I can find out myself, the White Horde had more relevance to Chinese/Mongolian affairs, but were relatively peaceful and kept to themselves, while the Blue Horde were the ones raiding Eastern Europe in the late 13th/early 14th century. I can’t find a source that properly explains this, or any interaction between the two separate factions.
AFAIK The most popular hypothesis is the division of Jochid Ulus (horde) between the White and the Blue Hordes/ Wings had root in the brothers of Jochi (d. 1227), Orda and Batu. While there are some significant confusion among the relevant sources on which wing is white/ blue, the majority of the scholars now seem to agree that the White Horde/ Wing (ak orda) was originally Batu's horde , and that the Blue Horde/ Wing (kök orda) was originally his senior brother, Orda's (Favereau 2021: 17f.; May 2018: 282; Jackson 2017: 182). As the Mongol Empire did for the Toluids' (descendants of youngest son of Chinggis Khan, Tolui) eminence, the Ordaids generally respected the domestic and diplomatic decision of the Batuids since they prioritized the political unity of the Whole Golden (thus Jochid) Horde, possibly due to the original succession priority.
(Added): >the White Horde had more relevance to Chinese/Mongolian affairs
This trend might also be relevant to the original superiority of the Batuid (White Horde) of the two in diplomatic matters.
May's annotation on the possible background of the confusion of the color/ wings (a few later Persian sources' negligence and the imbalanced focus of the Russian source on the western (right) area of the whole Golden Horde) might be useful to consider the historiography of this topic (May 2018: 283). [Kawaguchi & Nagamine 2016] also includes the detailed historiographical discussion of the division found in different sources.
Favereau's latest overview book on the Golden Horde approach the background of political conflict between both hordes (wings) from different economic interests especially surrounding the Black Sea/ Crimean Peninsula as well as the northern steppe route (Fevereau 2021: 190-92). The rivalry between the Golden Horde and the Ilkhanate as well as that between Genoa and Venice in the Black Sea primarily concerned the economic significance of the Black Sea/ Crimea as a trading hub in the commercial networks in the Eastern Mediterranean and the Middle East (that was also connected to Mamluk Egypt). It is also worth noting that the Golden Horde's expansionist policy in Eastern Europe in late 13th century is best represented by the interest of quasi-independent general Noghai (Nogay), whose power base was located in the steppe between the Don and the Dnieper (Dnipro) rivers (May 2018: 289-96). How Noghai (Nogay), ex-right-handed general of Khan Berke, consolidated his power by utilizing various diplomatic ties, has also attracted more attention in some recent studies, so to read some more on him might also shed better light on the understanding of the surroundings on OP's question.
I'm afraid OP might know much better than I (who've just only conducted a very basic research for this week's lecture in an undergraduate introductory course), but what I summarize above is the most basic line to approach OP's question, I suppose.
References:
(Edited): corrects typo.