Ok, do you have an extra large HDD you can use for this process? It will be easier to use this 3rd HDD to get disk images of both the laptop and desktop drives before you swap them. It is possible with just the 2 drives if you have enough free space on one to store an image of the other, but there is always the possibility of accidentally losing all the data on one of the drives this way.
There is also the licensing issues with Windows 10. Since I’m assuming the desktop is a store bought system, it has a Windows license that is bound to its hardware, specifically the motherboard. There is a chance that as long as they are both running the same version of Windows 10 (Home I assume) that they will be able to activate on the new hardware, but Microsoft can get weird with their licensing and activation.
You’ll also need software for this. Since you have a dock, I’m going to recommend Macrium Reflect Free. It’ll enable you to get an image of the laptop and desktop without taking the laptop apart. Clonezilla would also be an excellent choice if you are comfortable booting from a USB or CD.
With all this being said, I always recommend a clean install when you are switching users like this or doing a major change in hardware. I can understand your dad wanting to keep the computer “as is”, but your partner should consider doing a clean install and just move over what they need from the image of the other computer (docs, saved games, etc) and reinstall everything else. With such a drastic change in hardware, there is always the possibility weird things could happen and force you to clean install or fight with it down the line.