If you want the limiting distribution of the minimum of a collection of non-identically distributed chi-squares (each with a different degrees-of-freedom parameter), then this thread, Order statistics (e.g., minimum) of infinite collection of chi-square variates? I believe answers exactly the question.
If you want to see what happens to the distribution of the minimum of $n$ identically distributed chi-squares, as both the sample size and their common degrees-of-freedom parameter goes to infinity, then I have to offer the following remarks:
Assume then that $n/k \rightarrow 0$. It is well known that as $k$ increases the standardized chi-square approaches a standard normal distribution. Since in this case $k$ "goes to infinity faster than $n$", "at" infinity its standardized version is a standard normal distribution and its minimum should follow what the minimum of a normal distribution follows: using David & Nagarajah's notation, the CDF is
$$G^*_3(x) = 1-\exp{\{-e^x\}},\;\; -\infty < x < \infty$$
with $X$ being the minimum standardized by some $a_n, b_n$.
Note that $G^*_3(x) = 1 - G_3(-x)$ with $G_3()$ being the CDF of standard Gumbel distribution.
Now assume that $n/k \rightarrow \infty$. Here, $n$ escapes faster than $k$ and "at" infinity, the variable is -well, it is again a normal, because the normal approximation sets in much earlier than infinity. So it appears that we will again obtain the same result as in the $n/k \rightarrow 0$ case.