This could be a handy feature on the mobile version of Edge, would like to see this appear on other apps also

When it comes to mobile browsing, there are some who want no limits, including the background downloading of files. Although for many this option is of minimal concern, for others it is a highly requested feature. Therefore, questions about what Microsoft next-gen browser Microsoft Edge will and will not support is of vital interest.
Back in April, the Microsoft Edge team (known previously as Project Spartan), answered that background downloading was "not currently planned" for Windows 10 Mobile, but they were interested in feedback on the matter.
Today, that answer appears to have shifted to background downloading is in the works. However, you should not expect this feature immediately.