Nothing came with me when I moved, but I do end up in the US a fair bit thanks to work trips (and holidays to visit my awesome family).
Obviously I have really awesome things I love in the UK, but I have yummy things I have loved from various countries I have visited as well. If I find something I like, I'll bring it back if I have room in my luggage- because its fun to have yummy food at home from wherever I go!!!!

Things that make their way back with me from the USA are:
-Hot tamales. Mmm high fructose cinnamon corn syrup goodness....
-Local things New England things like maple butter, local apple sauces, (and although you can get it here obviously, the maple sugar farm by my parents house has amazing maple syrup and the local bee pollinators as well)
-Oyster crackers
-Goya products
-Zatarans
-Little Debbie Christmas Tree Cakes (when its that time of year)
-Tins of pumpkin, because I have no waitrose near me and I have never seen them anywhere else around here and I'm not that desperate to pay online £££££
-Big bags of Khona and Blue Mountain Coffee from Costco/BJS, because why not, its cheap compared to wee bags in the UK and tastes awesome
Also, my coworkers love Little Debbie Boston Creme Rolls and ask me all the time for them.