The policy in question is called closed-door operation, in which a bus that has crossed into a "foreign" jurisdiction is "closed" to passengers wishing to ride entirely within the foreign jurisdiction.
Westchester DOT operates closed-door within the Bronx, with the following exceptions:
- Routes 60/61 operate open-door along Boston Road (but not Fordham Road);
- Route 45 probably operates open-door within Pelham Bay Park. Note: I've never seen this; the two times I've ridden Route 45, nobody got on or off within the park.
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Similarly, MTA Long Island Bus operates closed-door outside Nassau County, with the following exceptions:
- Route N24 operates open-door along the section of Jamaica Avenue/Jericho Turnpike between 225th Street and 256th Street (where the center line forms the county border);
- Routes N31, N32, N33 operate open-door in Far Rockaway.
- Routes N19, N70, N72, N79, N95 operate open-door in Suffolk County.