I'm about to move to Toronto, ON, Canada, within months. I want to go ahead and start my business up now though, before I go there. Would there be any problem with this, and would I be able to operate my business when I move the same as if I were here in the States?
This completely depends on the type of business, and how easy it is to make sure you have an address where your LLC is registered.
As long as you can maintain a business address in the US (most UPS stores do something like this), then you can still maintain the business. Personally, I run multiple businesses based in the US, while living in the UK, where I also run multiple businesses. As long as you're willing to put in the time, and understand you might occasionally have to travel back to the states to address things, you should be OK.
If you want to provide more context, I'd be happy to give more input. Hope that helps.