The Constitutional issue concerning searches and seizures is the balance between a persons reasonable expectation of privacy, and the governments compelling need for actions to the contrary. Both are subjective at best, resulting in considerable litigation at the Supreme Court level, and decisions that apparently go both ways.
A Warrant is a court document which answers the question of reasonableness in advance. A warrant is proof that the agent involved has brought the case before a court and a judge has signed off on the existence of probable cause. Reasonableness can still be challenged when there is a warrant, but the likelihood of success drops significantly.
The bulk of litigation therefore involves warrantless searches and seizures.
To start, it is worth pointing out the distinction in wording between the fourth and fifth Amendments. The fourth states that the people shall be secure in their persons, while the fifth states that no person shall be deprived of due process. The fifth amendment has been consistently interpreted as applying to aliens, in the country legally or not. Interpretations of the fourth amendment have gone both ways. You will find commentators who state categorically that the fourth amendment covers aliens, but research will say otherwise. Best guess; if an alien has been in the country long enough to have become part of the local community, they will be protected, but as long as they remain just persons, they will not be.
That said, the next question regarding a warrantless search is location. The Court gives persons at border crossings much less expectation of privacy than people in their own homes. The practicality of a warrant in a border setting also swings the balance toward a warrantless search. There are still limits. Contrast the routine bag search and body scan at the airport, with being pulled out of line and directed to a interview room. The later reduces the possible discretion of the agent and allows for more formal grounds for suspicion.
While on the subject of borders, INS considers the border to extend 100 miles in from the actual line.
By contrast, the home, including garage and back yard is going to require a warrant, unless there is evidence of a crime or an emergency being in progress.
There is further information specific to warrants in the Appendix.