بسم الله الرحمن الرحيم
According to Shariah, it is not wajib for you to pay sadaqah al-Fitr of your adult children but if the children are sahib-e-nisab then it is mandatory for them to pay their sadaqah al-Fitr on their behalf. However, if the children stay with you in a joint family and you pay sadaqah al-Fitr on their behalf then there is no harm; their sadaqah will be paid. As when children stay with father then usually a father carries such tasks on behalf of his children. It is recorded in Dur Mukhtar with Shami: 3: 317, Zakariya:
…… لا عن زوجته وولده الكبير العاقل ولو أدى عنهما بلا إذن أجزأ استحساناً للإذن عادة أي لو في عياله وإلا فلا (در مختار) …. وعليه الفتوى (رد المحتار).
Allah knows Best!
Darul Uloom Deoband
This answer was collected from the official ifta website of Darul Uloom Deoband in India.