Moving data from old external drive (with old mac) to new external drive (and new mac)

I currently have an old MacBook Pro and I'm awaiting my new MacBook Air...I now use a Thunderbolt 2 external drive for my photos and music but want to now have them on a new Thunderbolt 3 drive. I'd rather not spend $50 for a converter for the old drive since I'll only need to use it once.

My question is this: if I hook up the old drive to the old computer and the new drive to the new computer and I'm using sharing mode, will I be able to transfer the files from the old drive to the new drive? (In other words--I know I can share from computer to computer, but will I be able to select the two drives in sharing mode? The external drives are much larger than my internal SSD drives.)

Thanks for any help!