If you receive a notice that either your insurance or taxes are due, call your new servicer and make sure that company has on file that funds have been escrowed for the premium. If the new company has not received a copy of that bill, it will probably direct you to send in the bill for payment. If you have a question after the transfer has taken place, you should contact your new servicer, even if your old servicer was the one that collected the funds for your insurance or tax payment.
Some mortgage companies offer to escrow life or disability insurance (insurance that would pay off the mortgage in case of death , or make payments in case disability). In these policies, the lender who originally made your loan is named as the beneficiary. If you have these policies, your old servicer should inform you of what effect the transfer of servicing will have on this insurance coverage and what action you may need to take to maintain coverage.
On flood and hazard insurance, it is the responsibility of the old servicer to provide the insurance agent or company with a notice of transfer. The beneficiary may be able to be transferred from one company to the other, but it is wise to make sure this occurs. You should make sure to transfer the beneficiary to ensure that, in case of a claim, the check is written and sent to appropriate servicer.