Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ The question is what is the relationship between the rank of $C_i$ and the rank of $A,B$? I conjucture that, for $i$ large enough, there exists an integer $s$ such that the following equality holds: $$\mathrm{rank}(C_i)=i\cdot\mathrm{rank}(A,B)+s.$$
$\mathbf{Edit:}$ Due to user1551's counterexample, the above conjucture is false. I think it should be revised as: for $i$ large enough, there exist non-negative integers $d,s$ such that the following equality holds: $$\mathrm{rank}(C_i)=id+s.$$
Your conjecture is false. E.g. suppose $$ A=\pmatrix{1&0\\ 0&0},\ B=\pmatrix{0&0\\ 1&0}. $$ Then $\operatorname{rank}(C_i)-i\operatorname{rank}(A,B)=(i+1)-2i=1-i$, which is not constant.