It is not easy for you to create a mount, so you should use it to its fullest. If you leave your mount spawned and unattended, it will slowly start deteriorating until it dies. Do not leave it unattended.
The way to de-spawn a mount works like this: aim it and walk up to it. Make it into a figure again by pressing alternate use( default G). after doing so, the figure will appear in your backpack.
Carry it with you whenever you go, if you use it. If you do not use it, you had better keep it safely in your bank, because others can loot it once you die.