I have LR Mobile running on my iPhone and syncing with a LR collection on my desktop.
I'd like to be able to copy the photos in the LR mobile collection onto my hard drive, but I can't figure out how to do this.
(essentially, I'd like to recreate what the iPhone photos app and cloud does with the Photos app on my mac)
I can't quite tell where you are in the process.
If you've set up synching, then the photos on your iPhone should show up in Lr's catalog on the desktop in "Mike's iPhone" in a side panel under "Imported photos." Those are on your desktop, as if it was a volume. Like any other photo in Lr you can move it around to some other folder in Lr.
For photos to be accessible to Lr Mobile app, you have to give it permission to access your camera roll. I think there's a setting for automatically including all the photos, but I can't recall. In any case you could create a collection, then add photos from the camera roll, then they'd sync to Lr on the desktop. Note that there's an irritation here, in that Lr doesn't recognize albums like you might have in iOS Photos, so you might wanna move stuff to collections in Lr Mobile, because those DO synch.
Also, it only syncs JPEG, not RAW. Which is a whole other topic.
And BTW, perhaps a better way to accomplish something Photos-like with your iPhone, but also integrated with Lr, is to try Mylio. It has much more effective and versatile synching than either Lr Mobile or Apple Photos, yet works with both, and can even pass Lr edits like cropping, exposure, etc, back and forth quickly. Even with RAW photos, and on stuff like removable drives, NAS, other computers, the cloud, etc. And unlike Photos you can selectively synch different