Vulgate: ut in nomine Jesu omne genu flectatur caelestium, terrestrium et infernorum,
Challoner: That in the name of Jesus every knee should bow, of those that are in heaven, on earth, and under the earth:
1582 Douay-Rheims NT: that in the name of IESVS euery knee bovve of the celestials, terrestrials and infernals
So, we can see both the Challoner revision and the original Douay-Rheims use “in the name”, as opposed to the “at the name” used in the KJV and the majority of subsequent translations. Possibly, the Latin “in nomine” is influencing this decision – the Douay-Rheims tends to try to make the English as close to the Latin as possible. (It is also worth noting “in nomine” is used heavily in the Catholicism’s Latin liturgies/rites, as part of the Trinitarian formula “in nomine Patris et Filii et Spiritus Sancti”, which in English is almost always translated “in the name of…” – and that liturgical use likely also influenced this translation decision).
This verse demonstrates the two main differences between the original Douay-Rheims and Challoner: antiquated spelling (IESVS instead of Jesus, euery instead of every, bovve instead of bow), and the original Douay-Rheims preferred Latinate vocabulary which in many (but not all) cases the Challoner drops, e.g. “the celestials” became “those that are in heaven”, “terrestrials” becomes “on earth”, “infernals” becomes “under the earth”.
The 1582 Douay-Rheims NT has copious footnotes. However, they don’t have much to do with the text, the ones on this verse are just attacking John Calvin and responding to various accusations he is claimed to have made against Catholicism. (They attribute these accusations to Calvin but don’t actually cite him, so I can’t judge whether they are presenting his views fairly.) This sort of old Catholic-vs-Protestant polemic is rather tiresome and boring, in my opinion, so I will say no more about it.