Not necessarily. It depends. I don't know if the estate is using a fiscal or calendar year. But for sake of discussion, let's say the estate's first year ended 12/31/2022. If there were no distributions made out of the estate before that date, then the estate will pay any taxes due on the computed taxable income.
Assuming the lump sum was in that tax year and the estate pays taxes on it, then when the distributions are made (going to another trust), any income say in 2022 would be passed into the trust's providence.
If the trust makes no distributions in 2022, it's taxable income would pay taxes. If the trust makes distributions to beneficiaries, then there may or may not be tax paid by the beneficiaries.
An amount of money distributed to a beneficiary does not necessarily equal (and probably never will) the amount of income the beneficiary may or may not pay tax on.
There are a lot computations, timing, etc. in this scenario that only someone with the full facts and amounts can plan out. What also comes into play are deductions. Legal fees, executor fees, etc., etc.
I would also recommend that in this complicated situation, you seek the advice of someone knowledgeable in estates and trusts. Either a tax attorney or CPA, enrolled agent, etc., that has knowledge in trust returns.
**Disclaimer: Effort has been made to offer correct information; but due to the discussion forum limitations, the poster disclaims any legal responsibility for the accuracy of the poster's response**
