With OMB's passback, the agency head has to decide whether or not to appeal the decisions - to OMB itself or even the President. The agency has to figure out what to do, and whether or not to "play ball" (and be a good boy or girl and make no waves), or to make noises. There are consequences either way, and many reasons for choosing either course of action.
Simply because OMB states what the agency is allowed does not mean that the agency head has to accept OMB's position. OMB may be bluffing, or OMB may not understand what is going on (for example, not fully representing the President's political position on an issue). The agency head may know better, and may decide to get back with the OMB director or even to appeal to the President.
The agency has to analyze the OMB passback and its implications, and decide whether or not there is value in appealing the passback. The appeal decision ultimately belongs to the agency head, and only the agency head can make the appeal to the President. Substitutes do not work on appeals.
An appeal can be based on a sense of fairness or justice, program's importance, prior decisions or precedents, new developments since the submission of the budget request, overlooked facts, and anything else that may help. It is usually a combination.
Operating component officials do not play a direct role in the appeals process. They may if they are responsible for a major element of what the agency head may be considering to appeal. The analysis of the passback may also involve various agency operating components.
After the passback, but must be done before final decisions are made on the budget by OMB or the President. (See To OMB, Timing for the timing of the decisions.)
Documents and Links
There are no public documents. The appeal itself may be a letter or memorandum, or may simply involve a few phone calls or even a meeting with the President. Prior to a meeting with the President there will be many briefing papers prepared, both by the agency and OMB.
Sometimes the press reports on what is going on. A good example of how the process plays out and its dynamics is in the December 8, 1998, issue of the Wall Street Journal: "Agencies Start Lobbying For Pet Projects in Budget" by Jacob M. Schlesinger and Bob Davis.