As in so many interesting questions, in the real world there are few simple answers, but a careful consideration of available practical options.
The factors that determine the possibilities and the correct course of action are rooted in the functions that have been assigned to local governments, the source of financing for those functions (untied, or tied transfers or own revenues) and the local economic situation. Poor local governments in low income countries, dependent on transfers from a central government that is itself in fiscal trouble may not have a choice of temporary remedies. At the other end of the spectrum, a local government with an extractive industry in its territory from which it receives substantial royalties may have more room to manuever than the central government. In between these extremes there are options such as postponing or slowing the expenditure on capital programs to temporary layoffs, rotations or shortened work weeks for staff.
If the local government has been wise and have consulted their constituents when preparing expenditure plans and programs, they will already have a reasonable sense of the community priorities. On this basis, they can prepare proposals to take back to the populace for confirmation. Naturally, not everyone will be happy with the choices but that seems in terms of results to be the best practice solution. This seems true even when taking degree of affluence into account. These issues are on the table at the moment in the USA and a reading of local news seems to confirm the argument here.
Local governments that have had good ongoing dialogue with their communites seem to be surviving the crisis with less political pain than those that have been less continuously open and forthright. The most recent example has been to adjoining local governments in my region. Both are quite affluent but one has been distinctly more open and prudent about realities on an ongoing basis. They also, not surprisingly, confronted the issues of adjustment earlier and are currently in less pain and substantially adjusted. The neighbour is facing considerably more turmoil.