Income earned by a deceased family member from January 1st through the day of their death will be reported on their final individual income tax return by April 15th the next year. Which means, that you won't know until the year is over, and tax reporting documents are mailed out by banks and employers, how much income was earned from January 1st through the day of their death. If the tax return results in taxes still being owed, the IRS will want that money. If the tax return results in a refund, the IRS will want to see proof that the correct relative is getting the refund according to a will or the laws of the state.
In Connecticut, the law orders estate proceeds to follow this order: children, (great)grandchildren, parents, siblings, nieces/nephews, (great)grand nieces/nephews, aunt/uncles, cousins, great(grand) cousins, etc. If there's no will and no relatives, then Connecticut gets the estate.
Income earned after the day of their death through December 31st will be reported on the estate income tax return; which is different from the estate tax return. [Generally, the estate tax return taxes the value of property that someone owns when they die. In 2023, you'll need to die with over $12.92 million to be hit with estate taxes in Connecticut.] The estate income tax return will only tax capital gains, while the remaining income will be taxed on the individual income tax returns for the estate beneficiaries (heirs/relatives). If the will requires that the estate to hold income until a grandchild graduates high school (for instance), then the estate income tax return will tax that income too.