Monotone circuit complexity of computing functions on sparse inputs
specifically considering one part of the question (eg for $k_1$=1,$k_2$=2), Lokam studied "2-slice" functions in this paper &amp; proves that strong lower bounds on them can be generalized, therefore this is a very hard open problem related to basic complexity class separation &amp; any such construction/explicit function would be a breakthrough; from the abstract:


  A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1’s and evaluates to one on inputs with more than two 1’s. On inputs with exactly two 1’s f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of 2-slice functions would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. 



Graph Complexity and Slice Functions / Satyanarayana V. Lokam,
Theory Comput. Systems 36, 71–88 (2003)


also as in his comments SJ covers this similar case in his book in the section  exploring star complexity of graphs sec1.7.2.


Boolean function complexity: advances and frontiers, Stasys Jukna