I can see two major drawbacks to an international aid approach - assuming it is towards a single cause or event response. First, it is very difficult to coordinate such efforts between different nations, especially on short notice. Each of the countries has a different amount and perhaps type of aid, being delivered by different means at different times and places. Some of the aid is secured and some of it needs securing, and there are language barriers. Secondly, foreign aid often comes with strings attached, as few countries give aid for completely altruistic (no self-interest involved) motives. So different nations attach different strings and often government sin crisis who need the aid are unwilling or unable to respond to the restrictions.
When you use this term, I assume that you are talking about having lots of countries get together to give foreign aid (rather than having individual countries give aid on their own).
If so, a major drawback is that there might be a lack of direction in the giving of aid. When countries are giving aid on their own, they tend to have specific agendas. They try to help the poorer country accomplish particular goals. With lots of countries trying to do aid together, they might have very different goals. So the aid might just be given in sort of a random way, touching on each country's goals a bit, but not enough to actually accomplish any of the goals. Also, less aid might be given out as the donor countries argued over what was best to do.
I do not see any inherent drawback in foreign aids as such, though individual foreign aids programs may have some drawbacks for donor nation, recipient nation, or both.
The term foreign aid gives the impression that foreign is some help given by one country or a group of countries to another country on purely altruistic considerations. But fact is that in all foreign aid programs both the donor and recipient country expect to benefit from. In reality, the benefit received or loss incurred by them will depend on the nature of the program.