Mishnah 7 in chapter 9 of Maseches Sotah seeks to clarify the status of the calf and the status of the murderer when the killer is found - whether during the process or after it.
"Nimtza ha'horeg ad shelo ne'erfah eglah":
If the murderer, who until that moment had not been identified, is found before the calf's neck was broken - "tetzei ve'tir'eh ba'eder" - she goes out and grazes with the herd. The calf returns to grazing with the rest of the herd, since she carries no sanctity whatsoever and there is no prohibition against deriving benefit from her. Even if she was already brought down to the rough valley, to that very place, she does not become forbidden until the actual breaking of the neck is carried out. Once the killer is found there is no longer any need for the eglah arufah, and she returns to her previous status.
"Mishe'ne'erfah eglah - tikaver bimkomah":
If the murderer is found after the calf's neck has already been broken - "tikaver bimkomah" - she is buried in her place, for it is impossible to derive any benefit from her, and this is the rule in the laws of eglah arufah: she requires burial. And the reason, in the words of the Mishnah:
"She'al safek ba'ah mitechilatah" - from the outset she was brought and her neck was broken because of a doubt, in a case where the identity of the killer is unknown.
"Kiprah sfeikah ve'halchah lah" - she accomplished what she was meant to accomplish, atoned for that doubt, and passed from the world.
Since she fulfilled her role and provided the atonement that needed to take place, the subsequent discovery of the murderer does not retroactively change her status: she accomplished her mission and remains forbidden.
The status of the murderer who is found after the neck was broken:
The Mishnah concludes that even though the eglah arufah completed her role and provided a certain atonement, she does not atone for the murderer: "Ne'erfah eglah ve'achar kach nimtza ha'horeg - harei zeh yehareg" - if the calf's neck was broken and afterward the killer was found, he is put to death. Even though the calf was already killed and atoned, judgment is still carried out against the murderer, provided that the proper laws of warning and witnesses and the like were fulfilled. The atonement that is granted is not for the murderer, but for the community and for whatever guilt it bears, and it has no bearing on the guilt of the murderer - and he receives his punishment.
In summary: We learned that finding the killer before the neck is broken removes the need for the calf, and she goes out and grazes with the herd; once her neck has been broken, she becomes forbidden to derive benefit from and requires burial in her place, since from the outset she came on account of a doubt, atoned for it, and passed on, and finding the killer does not retroactively change her status. We also learned that her atonement is directed toward the community and not toward the murderer, and therefore if the killer is found after the neck was broken, he is put to death.