Traditionally, the duty falls upon the eldest son (Karta). After marriage, he performs it along with his wife. If the eldest son is not available (deceased or otherwise unable), a younger son can perform it.
In the absence of a son, a grandson (son’s son) can perform the rites. If there are no sons or grandsons, the daughter’s son (dauhitra) holds a very significant right to perform Shradh. Even the brothers of the deceased can perform it if needed.
Regarding women: While the primary responsibility often lies with male descendants, women absolutely can perform Shradh, especially if there are no eligible male members in the immediate family or if they are unable to perform it. There are scriptural examples, like Mata Sita performing Pind Daan for King Dasharatha. In Kashi, we see daughters and wives performing these rites with full reverence and achieving the same merit for their ancestors. Unmarried women, however, are generally advised against performing certain specific elaborate rites like Tripindi Shradh alone, but can participate and perform simpler Shradh or Tarpan.