Rapid delivery systems where an order is delivered to a customer from a local distribution point within minutes or hours have experienced rapid growth recently and often rely on gig economy couriers. The prime example is a meal delivery system. During an operating day, couriers in such a system are used to deliver orders placed at different restaurants to different customer locations. Operating a rapid delivery network is challenging, primarily due to the high service expectations and the considerable uncertainty in both demand and delivery capacity. We seek to fill a gap in the literature by considering courier satisfaction in a rapid delivery system, which may improve retention/loyalty in a highly competitive environment. Under the premise that couriers prefer to operate in relatively small geographic areas to increase their efficiency, we propose the novel concept of dynamic courier regions: small operating regions for couriers which can be dynamically and temporarily expanded to allow delivery capacity to be shared between neighboring regions when necessary to keep customer service performance metrics high. We propose an optimization-based rolling horizon algorithm for courier management which handles both region resizing and delivery task assignment decisions. Experimental results for realistic settings demonstrate that the proposed algorithm successfully balances customer and courier satisfaction, simultaneously achieving delivery times that are comparable to those of a single operating region and courier satisfaction metrics that are comparable to those achieved by fixed, inflexible regions.