Once foreign aid is granted what is the process

What sort of foreign aid are you talking about here?

Are you talking about cash disbursements? Agricultural aid? Military aid? etc... They all have different processes.

I'm interested in particular how the workings of transfer of funds to other nations is done. Similar to the Marshall Plan at the end of WWII. Once congress approves the loans what are the actual mechanics of moving the monies from the US to the various countries? Is it through international banks or ??? Afraid I don't have enough knowledge in that area to ask the right questions

From what I can gather of the American system:

When Congress voted the international aid, the Federal Reserve bank is notified.

The amount, according to modalities voted, is tranferred (international funds transfer) to the national bank of the country aided.

This is done via an international organisation like the International Monetary Fund, the World Bank or others.

It's an electronic transfer which means it's not a hard cash transfer.

At agreed dates, a chamber of compensation or clearinghouse will determine the difference due between countries...

The individual bill passed by the Congress has the wording that says how processing is done. If could be done as Francis described for cash aid.

For military aid the Congress may approve a dollar amount and the other country establishes a Military Aid function with the State Dept. The State Dept works with the DoD and arranges for the other country to purchase what they want/need/desire. The country never gets the money. The money gets appropriated to the State Dept. by the Congress and they act an an intermediary. The country gets "goods" and the State Dept. reimburses the Dept. of Defense for the cost.

I believe agricultural aid works in a similar fashion but with the Dept. of agriculture instead of the Dept. of Defense.

In these latter two cases the other country never actually gets any money. They get goods delivered "in-kind".