The only supported way to keep photos libraries on external volumes, is to use a "sneaker drive", a portable drive, that you plug into the Mac where you want work with the library. Photos for Mac does not support the remote access to a photos library. You cannot keep the the Photos Library on a NAS, or a network share, or on iCloud Drive, Google Drive, or a DropBox. If you want use the same Photos Library on several Macs, the only supported way is iCloud Photos or the portable external drive.
See: Move your Photos library to save space on your Mac - Apple Support
If you want o merge your Photos Libraries from the three separate Macs into one library, you have several options, none is really easy - it will take a lot of time for large libraries: Notes on Merging Photos Libraries, 2019 Version