In the future, I would like to move around often-- say every 2-3 years. How difficult would it be to do this and have a career as an engineer?

It very much depends on where and what you area of engineering you would like to get into.  

Obviously, public sector would require staying within the area you serve. For example a water resources engineer for the state of California, could change to different regions but they would most likely be limited to open positions and where offices are located.  

Private sector in general would have a lot more flexibility, but would depend on where projects and/or offices are.   I have a friend working in private sector at a structural engineering firm, who moves often, and goes where the projects are. He currently is New Zealand.  

I would recommend first finding your specialty and look at firms that have multiple locations.  

Good Luck
Gillian