I honestly apologize for this long post

Insurance companies can't actually reduce your rate to match another company. That's called rebating which is illegal. You would have to do that for every single person. Unless the State's DOI (Department of Insurance) had stipulations on when a company could actually do something like truly match the rate. What they can do is a few things:
1) The insurance company I worked for in RARE instances would give a person credit on their account. This did not actually change their insurance premium but would mean so this time around they wouldn't owe as much. Let me tell you that was the worst thing we did because too many people expect their rate to actually be lower and it wasn't.
2) The can adjust your coverages to get you closer to what your other quote is which may or may not mean you have less coverage on your policy in order to get that. When comparing rates between companies you really have to compare apples to apples not apples to oranges and that is something virtually no one gets (i.e "this company is $100 less than yours why..well that's because you have BIPD coverage at 25/50 with that company but with us you have 100/300" just as an example)
3) Move you to another one of their products which may lower your rate
4) Add discounts to your policy that you are eligible for
5) Re-run your credit score. This however is regulated by the State in how often you can do this.
I'm probably missing a few other ways.
There isn't a problem with switching but do keep a few things in mind:
1) Initial rates for switching do not in any way reflect what your rates will be down the line. You may have a low rate now but that may not be true over several renewals. While this could be part of diminishing percentage of certain discounts it can also be due to the insurance company's claim ratio (it's common for people to not understand insurance is a pool of money therefore events happening across the county can impact your rate) or their need for a rate increase for other reasons.
2) Switching too often can result in a negative Insurance Score. Insurance Scores consist of your credit (CA is pretty much the only state where the DOI prevents factoring one's credit score into insurance rate), your past coverages as well as having insurance (with or without a lapse) over a pre-determined amount of time (usually with auto it's 6 months to 12 months), your claim and driving history. These are regulated by each state's DOI. Some states have rules on which claims or driving history can impact your rate and some states say you can't use a person's history of insurance (with or without a lapse). This could be important if for say you just got a speeding ticket or you filed a claim.