I read a news item about those GCT-30th Street Station trains in an old
Railroad Gazette at the New York Public Library. It was the announcement of the service. Central called them "Circuit trains."
Gazette, which was located in Manhattan, speculated that with the growing street traffic it might be faster to go from the East Side near 42nd Street to the West Side near 30th by way of Spuyten Duyvil than by horsecar direct. I think the idea was the city had pushed Central to do it.
Back in the day the old 30th Street station was the terminal for the Hudson River Railroad. After the 1869 merger (with the original New York Central) the Hudson River trains were moved to the new Grand Central Depot when it opened in October 1871.
Commuters in Yonkers in particular complained they could get 'downtown' quicker via the West Side line to 30th Street and a 9th Avenue horsecar than via the new route to Grand Central and the horsecars of the 4th Avenue line. If you have access to a
New York Times data base you can find some of the letters they wrote.
Some Yonkers-West Side through service was maintained but only until about 1890 I think. I thought it might've been later but the Yonkers newspapers carried the local schedules and one I looked at (on microfilm) from 1891 showed no through trains, only connections at Spuyten Duyvil. The connections between Hudson Division locals and West Side 'Dolly Vardens' were maintained at Spuyten Duyvil until the end of West Side passenger service (which I think we established in another thread was at least until 1930 or so).
Would make my life a lot simpler if they still had it today!