There is another option, to take on things yourself and grow beyond what is asked off you. If you just do what comes to you, you will loose your edge.
I'm pretty much sure that there would be a ton of things that you can do beyond your designated work.
Staying in the same organisation is not bad as long as you're not just staying where you are.
And while you stay longer, you will be able to get deeper into things and can influence org wide decisions, which would be relatively very difficult if you switch ever 2-3 years.