When you sign up to a new insurer you typically get various discounts - first year discount, online discount, etc. Sometimes there can be significant, sometimes even up to 30%. When it comes to renewal time these prices go up to standard.

Is there any downside to changing insurers every year to get the best price?

If course you want to use good companies - for car insurance I'm looking at state (incumbent), AMI, AA, and Tower. Tower's the cheapest by a reasonable margin for my car - $720 for state, $520 for tower, same excess and same type of policy. No idea about the detail of the cover.

Consumer customer satisfaction puts AA Insurance way out in front, State and AMI a little below average (they've been fine for me), Tower the bottom by a fair margin. I guess that's why Tower is cheaper

I guess if you're with an insurer for years they might be a little nicer during claims. I suspect though insurers will find any way they can to deny paying the claim or reduce the claim value regardless of the time you've insured with them.

I may also try calling my current insurer to see if they'll match the price before I change companies. I think I did that once before, they pointed out all the ways they were better but can't remember if they offered a discount or not.