If you need to just move one person's future assignments, use the 3 dot menu to access "Select All to the Right" and then you can drag them forward:
If you need to do this for several people (or several projects for one person), use the 3 dot menu to enable "Enable multi-select", and then select all the assigments and drag them forward. If you have some assignments that need to be split first, you can use the 3 dot menu to snip it before enabling multi-select: