If I leave my financial advisor will my new advisor handle my annuity?

This is a good question.

I'm going to assume that you already own an annuity and that your current financial advisor handled the transaction for this annuity sale.

If this is the case, it is likely that your current financial advisor has already received much of the compensation from the annuity sale.  

As a result, it's possible that there will be little financial incentive for your new financial advisor to handle or service your existing annuity.

This is a bit of a cynical view as it assumes that financial compensation is the only motivating factor for your new financial advisor.  

Does your new financial advisor work on a fee-basis?  If so, this would help avoid the potential conflicts of interest described in the scenario above.

Also, some annuities come with "trailing" commissions which continue to provide advisor compensation over the life of the annuity contract. Trailing commissions could provide some compensation for the new advisors' services if you make a "broker of record" change and assign the existing annuity to your new financial advisor.

Last, if both your old financial advisor and new advisor work primarily on a commission basis, you should be mindful of recommendations to change or "exchange" into a new annuity with your new advisor.  There might be a legitimate case to be made for an exchange of your old annuity to a new annuity through a Section 1035 exchange.  However, a recommendation to change by the new advisor could present a conflict of interest as the basis for the recommendation could be based almost entirely on the desire to receive compensation from the sale.

In any event, the net-net here is that you should ask whether your new advisor is working on a fee basis.  If this is the case, many of the issues and potential conflicts around bringing the existing annuity along could be avoided.