Lynx are found all across Middle-earth.
They will fight if threatened, providing a warning for you to back away before they attack. Not all are hostile.
Lynx can travel a long way, as Willem Whisker has demonstrated when escorting a certain Hobbit. Other lynx have become involved with quests and tasks or adventurers.
A listing of known Lynx extracted from the journals of travelers, and likely to be dangerous: