Confirm you have the request
Name the booking and the time they want to move, so they know you have the right one.
Check your notice rules
Look at your reschedule window. Inside it, move the booking; outside it, explain the fee or limit kindly and offer what you can.
Offer two or three slots
Give concrete times rather than asking when they are free. Fewer messages, faster answer.
Say what changes
Be plain about price, deposit or location changes so nothing surprises them later.
Confirm in writing
Send the new date and time in one line, and mention any reminder they will get.
Watch repeat movers
If one customer moves bookings often, add a gentle reminder of your policy at the next confirmation.