There is a help article about transferring files from TurboTax to ProSeries: https://proconnect.intuit.com/community/convert-data-files/help/how-to-transfer-turbotax-returns-to-...

I'm not certain about how to get the file from a remote client. If you can find the particular file, I would think you should be able to email it to yourself or if you're doing Remote Desktop Connect, you might be able to do a Copy-Paste from the remote computer to your own desktop.

As for steps once you've obtained the file, you are correct. You'll have to transfer the 2019 TurboTax file into 2019 ProSeries first. Then you can transfer the 2019 ProSeries file to 2020 ProSeries.