It seems that after Lightroom Classic downloads a file, it's re-downloaded in Lightroom CC
Lightroom CC needs to sync changes as well on a real-time basis that why its kept syncing file to Lightroom mobile version.
Please share the screenshot, where you see Lightroom CC says Downloading files.
I'm past where I can take screen shots, but this is how you can reproduce it.
1. Migrate Lightroom Classic catalog to Lightroom CC
2. Let Lightroom CC Sync all assets to the cloud until it indicates "Synced and Backed Up"
3. Leave Lightroom CC open
4. Open Lightroom Classic and let it start syncing to the cloud
5. As Lightroom Classic starts downloading cloud assets Lightroom CC begins to re-download those same assets
6. Pausing the sync in Lightroom Classic (or closing Lightroom Classic) will allow Lightroom CC to catch up and get back to a "Synced and Backed Up"
7. Un-pausing (or reopening) Lightroom Classic will cause Lightroom CC to also restart downloading images.
8. Once Lightroom Classic is done syncing to the cloud, Lightroom CC will catch up again and return to a "Synced and Backed Up State"
Eventually they'll both become fully synced. Fortunately I have a 1Gbps internet connect because it was A LOT of data movement.