Some types of visa like B1 may not expire with the expiry of the holder's passport. An unexpired, endorsed visa in an expired passport may be presented for entry into the USA, as long as the visa itself has not been cancelled, is undamaged, is less than 10 years old and is presented with a valid non-expired passport, provided that both passports are for the same nationality.
^ LIPTON, Eric (15 December 2006). "Administration to Drop Effort to Track if Visitors Leave". The New York Times. Retrieved 25 April 2019. Efforts to determine whether visitors actually leave have faltered. Departure monitoring would help officials hunt for foreigners who have not left, if necessary. Domestic security officials say, however, it would be too expensive to conduct fingerprint or facial recognition scans for land departures.
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Some foreign visitors sometimes engage in what is known as a visa run: leaving a country—usually to a neighbouring country—for a short period just before the permitted length of stay expires, then returning to the first country to get a new entry stamp in order to extend their stay ("reset the clock"). Despite the name, a visa run is usually done with a passport that can be used for entry without a visa.
Once in the country, the validity period of a visa or authorised stay can often be extended for a fee at the discretion of immigration authorities. Overstaying a period of authorised stay given by the immigration officers is considered illegal immigration even if the visa validity period isn't over (i.e., for multiple entry visas) and a form of being "out of status" and the offender may be fined, prosecuted, deported, or even blacklisted from entering the country again.

H-3 trainees: The length of the proposed training program, plus up to ten days before and after the start and end dates. If the initial program was designed to last for a shorter period than this but has been continued, the employer may request an extension from USCIS, up to the maximum authorized stay, which is two years (or 18 months for a special education exchange program).
Note: Visa applicants must qualify on the basis of the applicant's residence and ties abroad, rather than assurances from U.S. family and friends. A letter of invitation or Affidavit of Support is not needed to apply for a visitor visa. If you choose to bring a letter of invitation or Affidavit of Support to your interview, please remember it is not one of the factors used in determining whether to issue or deny the visa.
Generally, a citizen of a foreign country who wishes to enter the United States must first obtain a visa, either a nonimmigrant visa for a temporary stay, or an immigrant visa for permanent residence. Visitor visas are nonimmigrant visas for persons who want to enter the United States temporarily for business (visa category B-1), for tourism (visa category B-2), or for a combination of both purposes (B-1/B-2).
A valid U.S. visa in an expired passport is still valid. Unless canceled or revoked, a visa is valid until its expiration date. If you have a valid visa in your expired passport, do not remove it from your expired passport. You may use your valid visa in your expired passport along with a new valid passport for travel and admission to the United States. 
The United States is an open society. Unlike many other countries, the United States does not impose internal controls on most visitors, such as registration with local authorities. Our immigration law requires consular officers to view every visa applicant as an intending immigrant until the applicant proves otherwise. In order to enjoy the privilege of unencumbered travel in the United States, you have a responsibility to prove you are going to return abroad before a visitor or student visa is issued.
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