Late May 2017 I'll be staying somewhere in Luoyang (not booked yet).

Hi, there! You can take a taxi from the downtown area to the grottoes at a cost of CNY30. The suggested visiting time is 4~5h.
After that, still take a taxi to White Horse Temple in 30min and the taxi fare is CNY60. 2h is enough to finish the visit there.
There is no direct bus from either of the spots to Shaolin Temple. You may hail a taxi back to Luoyang Bus Station. Then take a bus to Dengfeng and the bus stops at Shaolin Temple in the midway; it takes 2h to reach the destination. May help.

thx - sounds as if I cld do the grottoes & the white horse in 1 day but the Shaolin Temple wld take a 2nd day. If I do grottoes then white horse later same day is there a bus from White Horse back to LY? how long does it take & how late does it run?

Well, the White Horse Temple is just located at Baimasi Town, Luolong District. After touring around, you can take bus no.56 to go back to downtown area. The last schedule of this bus is around 21:00.