Algebras for endofunctors bridge the gap between functors acting on a category and structures defined in it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more crucially a morphism of algebras is a map $a\to b$ between their codomains making the evident diagram commute. This construction is intrinsically homotopical, so it makes good sense to loosen the restriction from just endofunctors.
Do you know of any writing about algebras for more general transfors? Seems that many 2-categorical results about algebras over monads, lax morphisms, et cetera, would enjoy a more full account of transfors and their algebras.