That depends a lot on the constitutional right. They're, generally, phrased as restrictions on the federal government (assumed to apply to state governments under incorporation post civil war).
There are a lot of times the government is limited even dealing with foreigners abroad (in legal theory anyways, ymmv in reality).
However, countries may, depending on their laws, choose to not let certain people in on conditions that would otherwise violate guarantees on freedom of speech etc.
However, you do have your constitutional rights at the border etc. There is an exception concerning searches.