You may want to use background transfers instead of foreground:
https://winscp.net/eng/docs/ui_queue

Option to move running foreground transfer to background is tracked here:
https://winscp.net/tracker/1095

When multiple tabs are opened in winscp, uploading / downloading files to/from one server(tab) renders the other tabs unusable. The file transfer popup blocks interactions with other tabs.

When multiple tabs are opened, the file transfer window should be minimized in order to be able to use other tabs and carry out actions.

In this case, the only way to work with multiple tabs is to open a new instance of winscp, which defeats the purpose of tabs.