Nice fact: Suppose f:X->Y is a map of schemes and K⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and K are reduced, then it follows that f factors through K. This is nice because it makes "factoring through" purely a consideration of the underlying topological spaces.

So now I'm wondering, to what extent does "reduced" allow us to think only terms of topological spaces? Suppose we weaken the assumption that K→Y is an inclusion. When can we say f factors through K? More precisely:

Suppose X,K are reduced schemes, f:X→Y and g:K→Y are scheme morphisms such that f factors through g in Top. When does f factor through g in Sch?

I know the answer is "not always", for example if X,K,Y are fields and X,K are incomparable field extensions of Y (in Ring^op). But does anyone know any positive results we can state here?