Insurance rates are based on risk.  There are many factors in the amount of risk you represent to the company that they factor in to arrive at a payment for you.  I will note beforehand that you're free to switch companies any time you want, and you can shop for a deal any time, so if a rate hike is unreasonable, just use a different company.

Think for a few minutes about what factors create risk:

1. How far you live from work
2. How densely populated your area is
3. Weather - not directly, perhaps, but it affects accident rates in a tremendous way.
   
4. Car theft rates - not only be region, but by KIND OF CAR per region, IE, Hondas might be stolen more in NYC, but Toyotas might be stolen more in Miami.  This is considered in rates you pay.
   
5. State laws and regulations regarding car insurance can greatly affect rates - some states even force insurance companies to charge certain rates (MA did this until recently).
   
6. Accident rates among that age group in your area.  Live in a college town?  If a higher percentage of 20 year olds in your area crash their cars, you're going to help pay for it!

There's nothing wrong or illegal with this.  Keeping a car in NYC is going to cost a lot more in terms of theft, crashes, and vandalism than it will ever cost anyone in Bangor, Maine.

But, even if you move from one similar place to another, there may be differences in the rates at which accidents/theft/damage occurs, and if that's the case, you'll end up paying for it.

I will reiterate - if you move, you really should examine your insurance options some and shop around.  There's a good chance that whatever was cheapest in your previous home town is no longer the cheapest due to the way the rates are calculated.

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