Amy,
I'm sorry you have to deal with a snafu like this.
I noticed that the content most relevant to the keyword on the page tends to be pushed down below what reads like "SEO copy". For example, on the page about moving from the UK to Adelaide, Australia the entire first screen's worth of content is general info about Adelaide, and you have to scroll down pretty far to get to anything specifically about the UK. Another example is on the country-level pages (e.g. /spain, /thailand) the "Top Cities" with their score-bar and relevant links are pushed down below several long paragraphs about the country. Perhaps your users are already aware of the basic information about the country if they want to move there, and the more visually-appealing and helpful "Top Cities" area should be moved above? If you're worried about the SEO ramifications of moving the text content down you could just try this out as a test on a few country landing pages.
Let's think about the visitor for a minute. If I searched for "Moving to Dallas from the UK" and landed on this page I would not be happy: http://www.movehub.com/usa/dallas/move-to-dallas-from-uk . I would be looking for information on how to move to Dallas from the UK, things like visa requirements, good Vs bad neighborhoods, cost of living, job market, etc... Instead, all I see on my screen is a lot of fluff copy about how Texas isn't all about oil production anymore, and sidebar links to content about international container shipping costs, and moving to cities in Australia and Canada. The site needs a more intelligent way to show relevant content and links on a per-page basis. Some examples: Instead of showing links to Adelaide, Brisbane, Perth, Sydney, Toronto... on that page, show the cities closest to Dallas, like San Antonio, Austin, Houston... Instead of showing the general "about Texas" type content at the top, move it down or get rid of it alltogether and show the content that was once hidden below the fold up top: Moving To Dallas from the UK, and Comparing Dallas Vs London. Put in some links to pages about things like visa requirements from the UK to the US, and some job-search assistance (top employers in Dallas?) and you'll have a much more useful page.
More specific to the problem you experienced, however, if the content is no longer on the page it may just take Google some time to recrawl all of the old URLs again and see the updated content without all of the "keyword stuffing" they may have misunderstood. My advise would be to refresh your XML sitmap with new lastmod dates and resubmit it to entice Google to recrawl the pages again and see that the excessive keyword use has been fixed.