Partial intersection of 10 sorted lists of integers
The simplest way is to first remove duplicates from the 10 lists, and then merge all of them. Any number appearing 3 times in the merged list also appears in 3 of the 10 sorted lists.

A more complicated way is using pointers as you indicate. Briefly, the pointers start at the beginning of the lists, and then you keep incrementing the pointer pointing at the minimal element. You can use a heap to implement detecting which pointer to advance.

If you have $m$ lists containing $n$ elements in total, the first way takes optimal time $O(n)$ but uses up space (or modifies the input), and the second takes time $O(n\log m)$ (the $\log m$ comes from the heap) but uses less auxiliary space.