For example, if someone in the time of the temple dedicated a cow for a Karban, and before they could bring it to be sacrificed it became pregnant (or impregnated another cow), would its calf have any special status?
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The child of a Shelamim is offered as a Shelamim according to the Chachamim, while R Eliezer prohibits doing so. All further descendants of a Shelamim's child are not offered.
All descendants of a Todah are offered as Todot but without bread.
All descendants of a Bechor or a Maaser are treated as a Bechor or Maaser.
All descendants of a Chattat must die.
All descendants of a Temurat Olah are offered as an Olah.
All descendants of a female animal that was separated as an Olah should be left to graze until they get a blemish and then sold with the proceeds going to extra Temple offerings. R Eliezer holds that they may be offered as Olot themselves.
All descendants of a Temurat Asham should be left to graze until they get a blemish and then sold with the proceeds going to extra Temple offerings. R Elazar says you can use the funds for a personal voluntary Olah offering. According to R Eliezer they must die.