Strawberries live in the roots during the winter, the tops die off. The only problem I've ever had with mine (in containers) was letting them get too dry in the winter. They may be dormant but do need a little moisture to survive.
I usually transplant in the spring, but am guessing if you get enough of the roots, they will transplant most anytime (just a guess).
Each spring I start a new batch of plants from the runners of my best producers. Pull up the stunted or very large rooted plants (they will get thick stems and thick roots and rarely produce well) and replace them.
There are many varieties though. In my experience the runners that creep out and root as volunteers far from the parent plant are often the best.