Your plan to compare the hash across several systems may or may not produce the result you want, based on the origin of the hash on each system.
Hashes are the output of a one-way mapping function that will always (in theory) produce the same result for an given input, and will not produce that same string for any other input. The issue is that there are many many hash algorithms, and some alter the input in a predictable manner before hashing (called "Salting the Hash"). Others use a variable number of "rounds" where the resultant hash is rehashed multiple times.
If the hashes were created with the same hash algorithm, and no machine-specific salt was used, then both systems would generate the same hash for the same password, and your plan will work.
If however, the mechanism to generate the hashes on each box are different, or they include a system-specific token for salt, then no, the same password would generate a different hash output on each system, so you could be looking at two different hashes and not know whether they were based on the same password input.
See here for details about how the linux PAM uses hashing in /etc/passwd and /etc/shadow: https://wiki.archlinux.org/index.php/SHA_password_hashes