According to my research, the Constitution doesn’t stipulate how Congress must exercise the power of the purse. So the process itself would be up to Congress.
Here’s a brief analysis and flow chart.
http://nationalpriorities.org/budget-basics/federal-budget-101/federal-budget-process/
Which obviously means, the house still has the constitutional power of the purse and the power to adjust, suspend or repeal prior provisions of law enacted previously via so-called future appropriatons bills. The congress of 2011/2012 cannot bind the spending in fiscal years 2014 and beyond.