As noted in the comments, you seem to be misusing your private key.
Your private key should only be held locally. That is, on the hard drive of the computer you're currently sitting at.
You mentioned that
the private key is located on several locations from where I upload/download data
I gather you connect to one of this servers, do some work, and then commit to github. You do not need to have your private key on the server in order to do this. What you need is agent forwarding to your local machine, where your private key is located.
Now to answer your actual question, ssh private key files are just that. When you setup a passphrase for it you are encrypting that file assymetrically with the given pass. If you want to change that passphrase you have to decrypt the file and encrypt it again. Of course,
ssh-keygen provides a convenient way to do both steps with a command. As per @raz's comment link:
ssh-keygen -p -f ~/.ssh/id_dsa
Since you have a key file in each server, you will have to do this for each one.
Again, you should only have only one copy of your private key, on your local machine. Keeping it in each server is a hassle to maintain, and could lead to the key being compromised (although it's still hard, you are increasing the chances of that happening).