computer question that might be too convoluted for google

[wurm]

I have a USB 3.0 SD Flash drive that I store all my specific program' working files and projects. Sometimes in the course of downloading files from the internet to work with, to save time I just download them to the Desktop, then move them over to the flash drive as needed. It seems to me that in the past, when I moved files like that, they actually moved them. Now it seems they copy them, leaving the originals all over my Desktop.

Is there a simple process I can do that would look at both my Desktop and the flash drive and delete anything from the Desktop that's already on the flash drive?

I'm probably not explaining this perfectly, but I'm hoping you know what I mean. Thanks.