Thanks for your response. After reading what you wrote, I looked a bit more into my setup and I think I found my problem.
Most of my video is shot on my iPhone. I occasionally use the iPad for video. When I have an internet connection, the videos are supposed to be shared between my devices, and from appearances, they do. Except it seems that it doesn't actually download the entire video to the device.
So what was happening is that I'd shoot a video on the iPhone. Through iCloud, it would attempt to share that video to my iPad. From my perspective, it looks like the video is on my iPad. But in actuality, it's not. What's on the iPad is just a thumbnail. So when I use my iPad in order to edit the video on LumaFusion, it tries to download the rest of the video from iCloud, but if I don't have a connection, it can't do it.
I've found a simple workaround. When I shoot video on my iPhone, I can send it to my iPad using AirDrop, and then the full video is stored on the iPad and I can use it in LumaFusion as expected.
I feel rather silly that I didn't realize this sooner. Thanks for explaining how it's supposed to work so I could sit down and investigate why my device wasn't working the same.
Whew... when I was having this problem, I was thinking this was going to be a deal breaker since I plan to do a lot of shooting/editing in the field where I won't always have an internet connection.