There is an element of this that is a natural part of growing up. Kids mature at different ages and when they start maturing socially and a different rate from friends it can create strain. What one girl considers fun, the other will no longer think is fun because it is something "younger kids" do. The more mature one, may be more interested in boys and conversation than pretend play and the like. The troublesome part is that it never happens so clearly and distinctly as to make a parent able to just blame that. They can be wanting the "grown up stuff" one day and good old-fashioned play they used to do the next. Fights never look like they are over these issues on the surface, but may stem from this anyway if you look more closely.
Then, just as a woman not quite old enough for menopause, but past the age of having children might have symptoms of being "peri-menopausal" girls who aren't actually having their period yet, can still have hormonal fluctuations that make them moody, super-sensitive etc. etc. (This happens with boys too btw). At this age, many kids start developing that sense of being mis-uderstood. This happens with friends and parents and stems from the fact that most of us are not really very good listeners. This is another part of the emotional state (of anyone at any age really) that can create even further misunderstanding and fights.
Adolescence can begin as early as fourth grade, but even if that isn't the exact cause here, drifting apart from friends as well as having drag out fights with friends are both likely things in your daughter's future. Rather than moving and trying to help her avoid the situation (or fix it), I'd suggest helping her learn some coping skills as well as conflict resolution skills (paraphrasing, summarizing, use of "I statements", being able to ask clarifying questions . . .)
Most importantly, don't try to "fix" stuff for her but be there for her by just listening to what she is going through and supporting her as the loving parent you are. Ask her what she thinks might "fix" the problem and then ask her what she thinks the ultimate consequence of any of her "fix" ideas might be. Start asking her to analyze her ideas for solutions. This will help her learn some of those conflict resolution skills as well as coping skills I mention above, but it will also ensure she knows you love, support and respect her and that you are there for her, but that you can't just fix everything either - another lesson that needs to come along with growing up. As she gets older, it will also help prevent her from feeling misunderstood by you as much as possible.
You might also share stories from your own childhood about fights with friends and overcoming those, and start reading some, "coming of age" books together that also introduce characters that have to deal with similar conflicts. Anne of Green Gables and Berverly Cleary books come to mind as great classics for this purpose, but there are a ton of them out there and I'm sure a children's librarian in your area can help you find just the right thing for your daughter, her interests and circumstances as well as reading level.