Statistical Distance Growth Given K Independent Copies
See the "inclusion-exclusion" Lemma 2.2 here
https://www.cs.bgu.ac.il/~karyeh/mark-conc2.pdf .
For distributions $p,q,p',q'$, we have
$$ 
||p\otimes q-p'\otimes q'||
\le
||p-p'||
+
||q-q'||
-
||p-p'||
\cdot
||q-q'||.
$$

From here, it immediately follows that if $p=p_1=p_2=\cdots=p_k$
and $q=q_1=p_2=\cdots=q_k$ and furthermore $||p-q||=d$,
then the TV between the corresponding products is upper-bounded by $1-(1-d)^k$. This is proved explicitly in Lemma 4.2 in the linked paper.