How does the local ecosystem handle ensuring all urls are updated to new host environment when migrating to new host?

Our site: agnetic.com deploys extensive use of nested layers (main page, popups, and footnotes that popup too via bigfoot) each with lots of links.

Samplings at: agnetic.com and Meaningful Change Icebreaker - negotiating a fast-changing world

I’d like to know what migration plugin works best in moving from current wordpress hosting environment into localwp environment on the way to wp engine environment.

thanks