This is just to flesh out the approach using inserters and equifiers discussed in the answers, in a way that doesn't quite go down to the level of diagram chasing. I fear, however, that some things implicit in this argument woule require a diagram chase to carefully check.
Also, there's something I still find mysterious: what is it about inserters and equifiers which makes it so there is a particular leg of the limit cone and particular elements of the diagram such that certain hypotheses on these elements in the diagram ensure the forgetful functor down the special leg creates limits? To put a finer point on it: can we give a better description of the class of restriction functors among limits which (under certain partial limit preservation conditions) create limits? A better description than "whatever can be built up from inserters and equifiers"?
Lemma: Let $F,G: C\rightrightarrows D$ be functors, and let $Ins(F,G)$ be the inserter. Then the forgetful functor $Ins(F,G) \to C$ creates any limits that $G$ preserves.
For the proof, note that in in general, if $(c',\phi'), (c,\phi) \in Ins(F,G)$, then
$$ Hom((c',\phi'), (c,\phi)) = Hom(c',c) \times_{Hom(Fc',Gc)^2} Hom(Fc',Gc)$$
where the pullback is over the diagonal map.
Proof: Consider a diagram $(c_i, F(c_i) \xrightarrow {\phi_i} G(c_i))_{i \in I}$ in $Ins(F,G)$ such that $G(\varprojlim_i c_i) = \varprojlim_i G(c_i)$. Then $(F(\varprojlim_i c_i) \to F(c_i) \xrightarrow{\phi_i} G(c_i))_{i \in I}$ is a cone, and so induces a map $\phi: F(\varprojlim_i c_i) \to \varprojlim_i G(c_i) = G(\varprojlim_i c_i)$. We claim that $(\varprojlim_i c_i , \phi)$ is a limit of our diagram. Indeed,
$$Hom((c',\phi'), (c,\phi)) = Hom(c',c) \times_{Hom(Fc',Gc)^2} Hom(Fc',Gc) \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \varprojlim_i Hom(c',c_i) \times_{\varprojlim_i Hom(Fc',Gc_i)^2} \varprojlim_i Hom(Fc',Gc_i) \\ \qquad \qquad \qquad \qquad \qquad \qquad = \varprojlim_i (Hom(c',c_i) \times_{Hom(Fc',Gc_i)^2} Hom(Fc',Gc_i)) \\ \qquad \quad = \varprojlim_i Hom((c',\phi'),(c_i,\phi_i))$$
where we have used that limits commute with limits.
Lemma: Let $\phi,\psi: F \rightrightarrows G : C \rightrightarrows D$ be a diagram of categories, and let $Eq(\phi,\psi)$ be its equifier. Then the full subcategory inclusion $Eq(\phi,\psi) \to C$ is closed under any limits preserved by $G$.
Proof: This boils down to the limit of equal morphisms being equal.