d-ary heapsort analysis
Inserting $n$ items to a heap takes $\Theta(n)$.


You probably mean that "building" a heap takes $\Theta(n)$.


  Each delete-min takes $\Theta(d \lg n/ \lg d)$, I was thinking the bound is $\Theta(n d \lg n / \lg d)$.


I agree with you.


  How do I prove that it is indeed a tight bound?


How tight do you want to prove it is? Do you want to show that the worst-case running time of $d$-ary heap-sort is $\Omega(n d \lg n / \lg d)$? If so, I tend to think it is indeed tight.

For a hint, this paper: The Analysis of Heapsort mentions that (in Abstract)


  The number of keys moved during $2$-ary heap-sort when sorting a random file of $n$ distinct elements is $n \lg n + O(n)$ in the worst case.


It even further proves that (Notice that it is for the best case)


  Theorem 1 (Page 7): Heap-sort ($2$-ary) requires at least $\frac{1}{2} n \lg n- O(n)$ data moves for any heap composed of distinct keys. 


Maybe you can adapt it to $d$-ary heap-sort.