In order to manually move data between Google Drive and Dropbox, it may be best to handle the transfer yourself using a spare computer that can be fairly dedicated to the transfer task. The steps below show a method for moving large amounts of data from one service to the other.
Note: This only transfers _files_ from Google Drive, and does not handle "Google-native" documents, like Google Docs or Google Sheets, nor does it handle Email or other Google data.
1. Locate a less-used, LAN-connected computer that can be set to run the transfer over as long a period as needed. Depending on the amount of data to transfer, it can take days.
2. Log into the machine with an administrative account, and install both the Google Drive client ("Drive for Desktop") and the Dropbox app onto the machine.
3. Install WinDirStat (on Windows) or Disk Inventory X (on Macs), and run an analysis of your Google Drive.
4. Determine the amount of spare hard drive space on the machine (the larger the better). That amount is the size of the 'chunks' of data that you can practically transfer at a time.
5. Select files and folders for your first transfer by size, up to your 'chunk size', and set them to "Available offline" so that local copies are pulled down from the Google Drive servers.
6. When the download has finished (all files have green checks), start the upload to Dropbox by dragging them from the Google Drive into the Dropbox folder.
7. Monitor the Dropbox sync. When all of those files have synced up to Dropbox, delete them from Google Driveselect the next batch of files to transfer.
8. Repeat the process from Step 4 with your next set of files and folders (next 'chunk') as needed.
9. Make a note of any sharing permissions that you have set on folders in Google Drive. Those permissions will need to be added back to Dropbox once the transfer is complete. Once all permissions have been re-assigned, you can proceed to delete the files from Google Drive.