A comprehensive approach to streaming backups for virtual machines (“VMs”) in a storage management system comprises improvements to the assignment of data agent proxies for VM secondary copy operations. New considerations in performing a VM streaming backup job include without limitation: determining and enforcing a system-wide per-proxy limit of concurrent data streams; generating an ordered priority list of the VMs to be backed up as a basis for choosing which proxies will back up the respective VM, though the illustrative system may not strictly adhere to the priority list based on further considerations; identifying a next available proxy based on data stream utilization at the proxy; and dynamically re-generating the priority list and re-evaluating considerations if some VMs become “stranded” due to a failure to be backed up. Secondary copy operations are distributed to proxies in ways that improve the chances of successfully completing VM streaming backups.