They cannot enter a home without probable cause or a warrant. And outside of an actual port of entry, say at a random checkpoint they set up, they can only check IDs and look into cars the same as regular police. To perform an actual search of a vehicle in those circumstances still requires a warrant. From your first link at the cbp.gov website:

>Border Patrol checkpoints do not give Border Patrol Agents carte blanche to automatically search persons and their vehicles, other than in the manner described above. To conduct a legal search under the Fourth Amendment, the agents must develop particularly probable cause to conduct a lawful search. Probable cause can be developed from agent observations, records checks, non-intrusive canine sniffs, and other established means. Motorist's may consent to a search but are not required to do so.