It is known that fully faithful functors are closed under pushouts in Cat. Are locally fully faithful 2-functors closed under 2-pushouts in the 2-category 2-Cat of 2-categories, strict 2-functors, and 2-natural transformations? I expect this to be true, but giving an explicit description of a 2-pushout is daunting. Is there a simpler way to reason to prove this by reasoning entirely locally (i.e. in the hom-categories)?

I expect the fully weak setting to be more difficult, but if it is known that locally fully faithful pseudofunctors are closed under pseudopushouts in a bicategory of bicategories, this would also answer my question.