LoRaWAN is a widely used wireless communication standard for the Internet of Things that enables the collection of measurement data from numerous monitoring applications (such as smart metering, pollution, and asset tracking). Most research work has been focused on the performance of LoRaWAN as a function of the uplink traffic, as this is the most common scenario. Nevertheless, downlink traffic is a fundamental building block of the LoRaWAN standard, and a crucial part of applications such as smart healthcare, where reliability is extremely important. In this paper, we study the impact of the downlink traffic and different scheduling policies on the performance of LoRaWAN. We investigate: (i) the impact of an optimal schedule for the downlink traffic, (ii) the choice of the physical layer parameters used to send downlinks to end-devices during the second reception window, and (iii) the choice of the gateway that sends the downlinks. Our results show that even when using an optimal schedule, the presence of downlink traffic reduces the capacity of a LoRaWAN network up to 20%. The most limiting factor is the gateway, due to its duty cycle and half-duplex characteristics.