מפרנסים עניי נכרים עם עניי ישראל is not an obligation, it is permission. The gemara is telling the gabaim of a tzedakah fund, to which Jewish donors have given on the understanding that their money will go to עניי ישראל and they will therefore get the reward for tzedaka, that if it happens, while they are distributing the money to עניי ישראל that עניי נכרים show up and get in line, they are permitted to give to them too, even though whoever donated that money did not fulfill the mitzvah and will not get the reward. And the reason this is allowed is darchei sholom. The reason it’s not stealing is that the donors know this in advance, and give on the understanding that their money may go for that purpose.
The gemara further says that if a nochri wishes to donate to the tzedaka fund, and it’s not possible to refuse the donation, then it should be given to עניי נכרים so that the donor will not get a mitzvah, unless the donor specifically stipulated that it should go only to עניי ישראל.