If a baby was brought from beyond the techum on Shabbos, would that baby be muktzeh God forbid, or would it be permitted to carry it because a living thing carries itself.
The laws of objects, animals, or people coming from beyond the "Techum" (Boundary) on Shabbos, do not apply to a living human baby.
1) A live baby is not an object or animal. It is a person. People are not in the category of "muktzeh" or "objects" unless they are dead or Halachically considered dead.
People who have traveled on Shabbos beyond 2000 amos (about 4000 feet) from their city, or who started Shabbos in a wilderness spot etc. are restricted from travelling further on that Shabbos. Sometimes this restriction is measured strictly in a radius of about 8 feet around the person's position. (Dalet Amos).
The question of a living thing carrying itself, has no relation to the laws of "Techum". The prohibition of carrying on Shabbos is a separate law. Techum applies even if the person was carried, or simply walked beyond the limit without being carried.
2) Even though people who violate the Techum have certain travelling restrictions placed on them, these rules apply to adults or minors who are old enough to be obligated in such Mitzvah observance. A baby is not expected to observe these laws since obviously it has no capacity to understand them and is not responsible etc.
Also, the restrictions of Techum are not limits on specific activities, but rather limits on further travel. So one is perfectly allowed to attend to the baby.
Furthermore, Halachah would not penalize an adult from attending to the baby, because that would render the baby helpless.
Halachah does not penalize babies who "transgress". :)
3) It is our obligation to protect and help a baby. The consideration to provide constant care for the child is primary in this case (and most cases in Halachah). A baby is certainly more deserving of leniency, since small things a baby needs may actually create life threatening situations.
Therefore, in general, one faced with a Halachic question (and a qualified Rabbi is not available) which would deprive a baby of its needs, should usually err on the side of caution and common sense, and grant the baby what it needs first, and then ask the Rabbi later when you find him.
However, one should learn the laws well and strive to have access to a qualified Rabbi.
4) Shulchan Aruch Orach Chaim 406:1 states that someone who unknowingly violated the Techum, may have their restrictions lifted and may travel if they need to seek a proper place to use as a bathroom.
The Kaf HaChaim comments here, that the allowance is not just for a bathroom but for shelter from the rain or if the person is uncomfortable in the heat etc.
His words are: "Great is the honor of people that a negative commandment is pushed aside."
The source is the Talmud Eiruvin 41b. The Sages of Nehardea also explain there, that a person who is restricted outside the Techum, should be clever and seek a bathroom spot close to the city area, and then once there, we allow him to come back since he is already there.
Also, see S.A. O.C. 404:1 with Rema. The Rema explains that one who arrived (and there it does not call the violation "unknowing") by boat on Shabbos, may leave the boat and enter the city due to various discomforts.
From this we can learn out by comparison, that a baby's discomfort is obviously grounds to lift any restrictions of "Techum".
Of Course, if the parents and baby have no discomfort and can fulfill all the laws of Techum without further danger or discomfort, then they should act normally as in any Halachic situation.
For further understanding of the Laws of Techum, see Shulchan Aruch Orach Chaim 396 - 416.
I hope this helps.