Operational processes in production, logistics, material handling, maintenance, etc., are supported by cyber-physical systems combining hardware and software components. As a result, the digital and the physical world are closely aligned, and it is possible to track operational processes in detail (e.g., using sensors). The abundance of event data generated by today's operational processes provides opportunities and challenges for process mining techniques supporting process discovery, performance analysis, and conformance checking. Using existing process mining tools, it is already possible to automatically discover process models and uncover performance and compliance problems. In the DFG-funded Cluster of Excellence "Internet of Production" (IoP), process mining is used to create "digital shadows" to improve a wide variety of operational processes. However, operational processes are dynamic, distributed, and complex. Driven by the challenges identified in the IoP cluster, we work on novel techniques for comparative process mining (comparing process variants for different products at different locations at different times), object-centric process mining (to handle processes involving different types of objects that interact), and forward-looking process mining (to explore "What if?" questions). By addressing these challenges, we aim to develop valuable "digital shadows" that can be used to remove operational friction.