It is often stated that eurodollar borrowing is clost substitute for Fed funds borrowing. In other words, when US banks cannot fund themselves domestically, they might go to the eurodollar market and borrow some money to satisfy thier reserve requirment. Although it seems intuitive that in eurodollar borrowing you are borrowing from banks in other countries, I believe it implicitly involve a loss of reserve in a US bank. This is because since foreign banks cannot hold reserve at Fed and the borrower wish to borrow reserve, the only way this can happen is for the foreign bank to demand its US correspondant (the US bank where it deposits its dollar) to transfer the amount of reserve to the borrower. So implicity, a US bank is losing its reserve. Am I correct?
I think you are right that ultimately all dollar movements are reflected in reserve accounts at the Federal Reserve.
May I make a couple of additional points: Eurodollar borrowing is really not a close substitute for Fed Funds. First of all, not much volume goes through the term eurodollar markets, so it doesn't have the capacity to replace Fed Funds borrowing. In a dire situation where a bank cannot obtain overnight loans in the Fed Funds market, I would think there's no chance it could borrow in the eurodollar market.
I would add that right now all this is moot, because we are in a situation of massive excess reserves due to Quantitative Easing.