The COVID-19 pandemic has placed severe demands on healthcare facilities across the world, and in several countries, makeshift COVID-19 centres have been operationalised to handle patient overflow. In developing countries such as India, the public healthcare system (PHS) is organised as a hierarchical network with patient flows from lower-tier primary health centres (PHC) to mid-tier community health centres (CHC) and downstream to district hospitals (DH). In this study, we demonstrate how a network-based modelling and simulation approach utilising generic modelling principles can (a) quantify the extent to which the existing facilities in the PHS can effectively cope with the forecasted COVID-19 caseload; and (b) inform decisions on capacity at makeshift COVID-19 Care Centres (CCC) to handle patient overflows. We apply the approach to an empirical study of a local PHS comprising ten PHCs, three CHCs, one DH and one makeshift CCC. Our work demonstrates how the generic modelling approach finds extensive use in the development of simulations of multi-tier facility networks that may contain multiple instances of generic simulation models of facilities at each network tier. Further, our work demonstrates how multi-tier healthcare facility network simulations can be leveraged for capacity planning in health crises.