Lemma 5.25.
In both cases, the evident map from the original span to the new span induces an equivalence on pushouts.
Proof
For the left construction, the new pushout is
\[Q^{\ell} \sqcup_{P^{\ell}} R^{\ell}
\simeq
Q \sqcup_{P^{\ell}} (P^{\ell} \sqcup_P R)
\simeq
Q \sqcup_P R,\]
by pasting of pushout squares. The right construction is symmetric.