When you file an over 65 homestead exemption your school and county taxes are capped at the current tax amount. This means as long as you own the property, you will not pay more in taxes (school and county only) than the year you filed the over 65 homestead exemption. All the other entities are not subject to this capped amount, so your overall tax bill can go up (i.e. your city taxes increasing).
So what happens you move? Your cap amounts do not transfer, but rather the percentage of taxes you are paying transfers. Let's say your school taxes are capped at $5,000 and uncapped they are $10,000. This means you are paying 50% of the uncapped amount. The 50% is what transfers over, not the $5,000. So if your new property has an uncapped school tax amount of $12,000, your new school tax cap would be $6,000 ($12,000 * 50%).