"Relatively free" functors have been considered at least since the time of <a href="http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf">Lawvere's thesis</a>; see for example page 111 of 122 for the case involving finitary algebraic theories. 

I don't know where this is written down, but the following construction is pretty general and might suit your purposes. Let $\theta: S \to T$ be a morphism of monads on a category $C$, and suppose that the category of algebras $C^T$ has coequalizers (as happens if for example $C= Set$). Then the forgetful functor 

$$C^T \to C^S,$$ 

which takes a $T$-algebra $(d, \alpha: Td \to d)$ to the $S$-algebra 

$$Sd \stackrel{\theta d}{\to} Td \stackrel{\alpha}{\to} d,$$ 

has a left adjoint. This gives the factorization in the case where $\mathcal{B} = C^S$ and $\mathcal{C} = C^T$. 

The left adjoint takes an $S$-algebra $(c, \xi: Sc \to c)$ to the coequalizer of the pair of arrows in $C^T$:

$$T\xi: TSc \to Tc, \qquad TSc \stackrel{T\theta c}{\to} TTC \stackrel{mc}{\to} Tc,$$

where $m: TT \to T$ is the multiplication on $T$. This coequalizer might be written $T \circ_S c$, by analogy with the coequalizer $B \otimes_A M$ where $B$ is an algebra over $A$ and $M$ is an $A$-module. 

That this is the left adjoint is a slightly lengthy diagram chase which I can produce if desired.