Process mining techniques can be used to analyse business processes using the data logged during their execution. These techniques are leveraged in a wide range of domains, including healthcare, where it focuses mainly on the analysis of diagnostic, treatment, and organisational processes. Despite the huge amount of data generated in hospitals by staff and machinery involved in healthcare processes, there is no evidence of a systematic uptake of process mining beyond targeted case studies in a research context. When developing and using process mining in healthcare, distinguishing characteristics of healthcare processes such as their variability and patient-centred focus require targeted attention. Against this background, the Process-Oriented Data Science in Healthcare Alliance has been established to propagate the research and application of techniques targeting the data-driven improvement of healthcare processes. This paper, an initiative of the alliance, presents the distinguishing characteristics of the healthcare domain that need to be considered to successfully use process mining, as well as open challenges that need to be addressed by the community in the future.