Much of this answer is developed at length in a teshuva in BeMareh HaBazak IX:6.[1] We will add points that apply specifically to your case.
There are two halachic issues to consider: commerce on Shabbat and amira l’nochri (requesting a non-Jew to do work on Shabbat).
Commerce: Rashi[2] gives two possible reasons for the prohibition of commerce on Shabbat. First, conducting commerce violates the navi’s instructions to refrain from mundane activities; second, it may bring one to write. Simple logic dictates that when the Jew’s involvement in the commerce was complete before Shabbat began and matters are finished by a non-Jew or automatically, the prohibition should not apply, since neither of these two reasons are pertinent. However, Rabbi Akiva Eiger[3] extends the prohibition of commerce to include cases in which the interaction took place during the week if it was set up to take effect on Shabbat. One can therefore argue that regardless of when you were active in setting up the future payments, it is forbidden to arrange matters so that the payment of your debt (which constitutes commerce[4]) will take place on Shabbat.
Nevertheless, one need not be concerned by this opinion for a combination of two reasons: 1) It is far from clear that we accept Rabbi Akiva Eiger’s opinion (see opinions in our teshuva in BeMareh HaBazak). After all, even beginning full melachot before Shabbat that finish by themselves on Shabbat is permitted. Logically, we would expect that the Rabbinic prohibition on commercial activity on Shabbat would not be more stringent. 2) There are accepted sources[5] that a Jew may have a non-Jew acquire something on his behalf on Shabbat.
It is possible to argue that Rabbi Akiva Eiger’s thesis was that a transaction finished by a non-Jew is permitted on Shabbat, while one that is finished by itself is forbidden.[6] There are additional distinctions between this case and that of Rabbi Akiva Eiger as well.[7] Therefore, it is likely that Rabbi Akiva Eiger would agree that your case, in which a Jew gives over before Shabbat the process to be finished by a non-Jewish financial institution, is permitted.
Amira l’nochri: The main reason to be lenient is technical. Our research indicates that there is no need for human intervention at the time of the transfer of funds.[8] Since the transfer could be finished before Shabbat, even if the non-Jew chooses to do it on Shabbat, the Jew faces no halachic problem.[9] If the bank takes a set fee per transfer (katzatz), we consider it as though they are acting for their own benefit,[10] and the permissibility is even clearer. Although it is true that even in a situation of katzatz, a Jew may not tell a non-Jew to perform the act specifically on Shabbat, in this case, even if a melacha would have been needed at the time of the transfer, it likely could be done after nightfall of Motza’ei Shabbat, or probably even a day later or earlier (with the account charged afterwards).
We could stop here, but we do not want to imply that there are no other potential grounds for leniency. Amira l’nochri is among the most complicated areas of the laws of Shabbat, and people should become accustomed to asking halachic questions in this area, as there may be more room for leniency or stringency than one might expect.
We should consider whether the bank is working for you when making the transfer. Alternatively, perhaps the transaction between the bank and the credit card company is being done primarily on behalf of the credit card company. The credit card company is the one who initiates the payment on a monthly basis, when the time comes, and it receives the money. Perhaps we should therefore view your instructions to the bank as your acquiescing in advance to the payment (although if it were not worth your while, you would not do it). Our conclusion might differ from case to similar-sounding case.
It is also possible that giving the order during the week to pay on a day of the month that usually falls on a weekday is not considered amira l’nochri, even if, down the line, it happens to fall out occasionally on Shabbat or Yom Tov. There are several precedents for the idea that statements that cause a non-Jew to work on Shabbat may still not be considered direct enough to be amira l’nochri.[11] However, it is very difficult to apply such a concept.[12]
The bottom line is that you may allow this bank transfer arrangement to continue no matter when the transfers actually fall out.