Not every U.S. citizen wishing to obtain a green card for a foreign-born family member actually lives in the United States. Some, for example, get married while living overseas, and some are members of families that are based abroad. At some point, however, the U.S. citizen and immediate family might wish to move to the United States.
This raises the question: Can an overseas U.S. citizen sponsor family members for U.S. green cards (U.S. lawful permanent residence) allowing them all to move to the U.S., together or apart? It's possible, but it won't be straightforward, as described below.
First, understand that even a U.S. citizen cannot petition for (sponsor) all categories of family members for a U.S. green card. The only relations who are allowed immediate admission (without spending years on a waiting list) are immediate relatives of U.S. citizens; that is, parents, a spouse, and unmarried minor children (including stepchildren if their parents' marriage took place before the child turned 18).
A U.S. citizen can petition for certain other family members (adult children, married children, and siblings), but the wait for their "priority date" to become current could be long, depending on category and which country they're from. See Green Card Through a U.S. Family Member: Who Qualifies? for details.
To start the process of obtaining green cards for family, the U.S. citizen will need to prepare and submit USCIS Form I-130 to the appropriate U.S. government agency.
See, for example, Preparing an I-130 Visa Petition for the Immigrating Spouse of U.S. Citizen or Filling Out and Submitting Form I-130 for Parents of a U.S. Citizen or Filling Out, Submitting Form I-130 for Unmarried, Minor Child of U.S. Citizen or Permanent Resident.
After that petition is approved, the immigrating family members themselves would need to prepare forms and documents, and eventually attend an interview at a U.S. embassy or consulate in their country of residence. (This is part of what's known as "consular processing.")
Here is where things can get sticky. All intending immigrants must prove that they are not "inadmissible" to the United States. U.S. immigration law contains numerous grounds of inadmissibility, which bar noncitizens from entry for security, criminal, health, financial, and other reasons.
Specifically, applicants are expected to prove that they will not likely become a "public charge" (dependent on need-based government assistance). To prove this, the U.S. petitioner must sign paperwork promising to act as financial sponsor in the United States. The primary paperwork in question is a Form I-864 Affidavit of Support.
But only someone who is "domiciled" in the U.S. (living there) is eligible to serve as a sponsor. A U.S. citizen who doesn't plan to move to the U.S. can (and must, by law) still fill out a Form I-864, but it won't be enough to overcome this ground of inadmissibility. The U.S. citizen will most likely need to find someone within the U.S. to serve as a joint sponsor. This is no small favor to ask, as described in What Sponsors Should Know Before Signing Form I-864 Affidavit of Support.
If the U.S. citizen does plan to move to the U.S. with the rest of the family, this issue will be much easier to overcome. The U.S. citizen sponsor will need to provide evidence of these plans, however, ideally including a job offer in the United States (or sufficient assets to replace the need for a job).
It would be wise to consult an experienced U.S. immigration attorney, preferably in the state where the immigrating family members intend to live or where a likely joint sponsor resides.