The way I would handle this is tell your current IRA broker you want to close your account and receive a check. They send you the paperwork, you return it, they send you the check. From the date of the check, the clock is ticking. You deposit their check in your checking account and write your own checks to the new custodians. Voila! You MUST have the cash in the new IRAs within 60 days. If it takes you 61 days it is a premature distribution, subject to tax and penalties, and you can't put it in the new IRAs.
Now, this is called a rollover. You are allowed to do it once in a 12 month period. Your current custodian will report the disbursement as a distribution, premature if you are under 59 1/2, but it is OK, the new IRA custodians will report the "rollover contribution" and you will see to it that they add up, as will the IRS.
Note that this is different from the rules when a 401k does rollover to an IRA. Then you may not take the money, even for a second, or there is that tax levied--which you can get back when you file your taxes if you have indeed put the money in an IRA, but you have to pay taxes at an inconvenient time. But you tell us the money is already in an IRA, and you want to move it to several different ones.
Best wishes, Chris