The emerging need for subnational estimation of demographic and health indicators in low- and middle-income countries (LMICs) is driving a move from design-based methods to spatial and spatio-temporal approaches. The latter are model-based and overcome data sparsity by borrowing strength across space, time and covariates and can, in principle, be leveraged to create yearly fine-scale pixel level maps based on household surveys. However, typical implementations of the model-based approaches do not fully acknowledge the complex survey design, and do not enjoy the theoretical consistency of design-based approaches. We describe how spatial and spatio-temporal methods are currently used for small area estimation in the context of LMICs, highlight the key challenges that need to be overcome, and discuss a new approach, which is methodologically closer in spirit to small area estimation. The main discussion points are demonstrated through two case studies: spatial analysis of vaccination coverage in Nigeria based on the 2018 Demographic and Health Surveys (DHS) survey, and spatio-temporal analysis of neonatal mortality in Malawi based on 2010 and 2015--2016 DHS surveys. We discuss our key findings both generally and with an emphasis on the implications for popular approaches undertaken by industrial producers of subnational prevalence estimates.