Select in union of sorted arrays: Already known?
Frederickson and Johnson obtained an optimal result in the 80s. Let $p=\min(k,t)$, then there exist an algorithm solves your problem in $O(k+p \log \frac{t}{p})$. 

Reference

G.N. Frederickson, D.B. Johnson "The complexity of selection and ranking in x+y and matrices with sorted columns" 
J. Comput. System Sci., 24 (2) (1982), pp. 197–208