An IRA inherited by a non-spouse beneficiary can only be moved by nonreportable trustee-to-trustee transfer. A trustee-to-trustee transfer is neither a distribution nor a rollover.
A distribution paid to the beneficiary is not permitted to be deposited into another inherited IRA. It is taxable income to the beneficiary on the tax return for the year in which the distribution occurs. In TurboTax this is done by entering the Form 1099-R for that year that reports the distribution.
As a non-spouse beneficiary, once money is distributed to you from an inherited IRA it becomes simply cash. You can use this cash as you would any cash, including using it to fund your own IRA contribution if you qualify to make such a contribution, but it is not permitted to be deposited into an IRA as a rollover or Roth conversion. Only a spouse beneficiary is permitted to do either of those.
