Diagrams $F:\mathcal{C}\to\mathcal{D}$ are objects of the functor category $\mathcal{D}^\mathcal{C}$, when $\mathcal{C}$ is small. A particular case of a small category is the category associated to a directed poset $\Lambda$, where there exists a morphism $f:\beta\to\alpha$ if and only if $\alpha\leq\beta$. Diagrams $F:\Lambda\to\mathcal{D}$ are then objects of $\mathcal{D}^\Lambda$, which send some $\lambda\in\Lambda$ to some $D_\lambda$ in $\mathcal{D}$. The category $\mathcal{D}^\Lambda$ has natural transformations as morphisms, but let's consider natural isomorphisms $\eta:F\Rightarrow F$. Such a natural isomorphism $\eta$ consists of component isomorphisms $\eta_\lambda:D_\lambda\to D_\lambda$ such that for $\alpha\leq\beta$ and $f:\beta\to\alpha$ in $\Lambda$ we have $\eta_\alpha\circ F(f)=F(f)\circ\eta_\beta$.
I believe this corresponds to the case of an inverse system in $\mathcal{D}$: Indeed we have objects $D_\lambda$ for each $\lambda\in\Lambda$ and we have transition maps $\varphi_{\alpha\beta}:=F(f):D_\beta\to D_\alpha$ for all $\alpha\leq \beta$ in $\Lambda$. The inverse limit of this system is the limit of the diagram $\varprojlim_{\lambda\in\Lambda}D_\lambda$.
Is there a sense in which there is a corresponding limit $\varprojlim_{\lambda\in\Lambda}\eta_\lambda:\varprojlim_{\lambda\in\Lambda}D_\lambda\to\varprojlim_{\lambda\in\Lambda}D_\lambda$? How is this constructed? I am interested in this because automorphisms of $\varprojlim_{\lambda\in\Lambda}D_\lambda$ could be pretty pathological, swapping isomorphic factors of the diagram, etc. Considering natural isomorphisms of the factors would help me sort out a problem with the semidisjoint union in profinite quandle-land.
$\newcommand{\C}{\mathcal{C}}\newcommand{\D}{\mathcal{D}}$You seem to be discovering the fact that (co)limits are functors. In a suitable context. We don't need to reduce to automorphisms or posets, this is quite general.
Suppose $F,G\in\D^\C$ are two diagrams whose limits exist, and suppose $\eta:F\implies G$ is a natural transformation. I claim there is a canonical arrow associated to $\eta$, which we call $\varprojlim\eta$, from $\varprojlim F$ to $\varprojlim G$. When $\D$ admits all limits of shape $\C$, this assignment $\eta\mapsto\varprojlim\eta$ makes $\varprojlim:\D^\C\to\D$ a genuine functor.
To get an arrow $\varprojlim F\to\varprojlim G$ is to specify a cone over $G$ with apex $\varprojlim F$. We define the legs of this cone, for $c\in\C$, to be: $$\varprojlim F\to F(c)\overset{\eta_c}{\longrightarrow}G(c)$$Where the first arrow is the leg of the limit cone. Exercise: use naturality of $\eta$ to show that this really does define a cone $\varprojlim F\implies G$.
The universal property of a limit induces from this cone a well-defined arrow $\varprojlim\eta:\varprojlim F\to\varprojlim G$.