I think what you are looking for is mixed codes.
A good start point would be Brouwer--Hämäläinen--Östergård--Sloane. They are talking about mixed binary/ternary code, so for some $k$, $n_1=\cdots=n_k=2$ while $n_{k+1}=\cdots=n_N=3$. Brouwer keep an online list of known 3/2 mixed code. I think they also talked about some general cases. Another interesting paper is Perkins--Sakhonivich--Smith.
Anyway, more papers can be found from the references therein or by the key word. I also find this online list with 4/3/2 mixed covering codes and many references.
update: Turbo mentioned a work of Lenstra in the comment. It already uses the term "mixed codes" on the first page.