This depends on which sacrifice and the status of the mother at the time of the Hekdesh.

In general the offspring of a Chatas cannot be brought nor redeemed. Rather it must die. A Chatas can only be brought for the sin for which it was originally designated. The offspring of the Chatas is one of five unneeded Chataos which must die. However, there is a Tanna that says that if the mother was already pregnant at the time of the Hekdesh then one may use either the mother or the child for his Chatas and the other can be redeemed.

An Olah or an Asham is male and therefore does not have offspring.

A Shelamim's offspring can be offered as a Shelamim

Dov Zupnik