Consider the problem of finding the limit of the following diagram:

$$ \require{AMScd} \begin{CD} & & & & E \\ & & & & @VVV \\ && C @>>> D \\ & & @VVV \\A @>>> B \end{CD} $$

The abstract definition of the limit involves an adjunction related to collapsing the entire index category to a point. However, one could break this operation into two stages: first collapsing the upper three objects to a point reduces it to

$$ \require{AMScd} \begin{CD} & & C \times_D E \\ & & @VVV \\A @>>> B \end{CD} $$

and then we finish computing the limit as $A \times_B (C \times_D E)$.

This is a particularly convenient thing, since it implies a way to work locally with more complicated diagrams where you ultimately want a limit — i.e. take limits or perform other modifications to smaller pieces of the diagram while leaving the rest unchanged.

However, not every variation works out so nicely. If we try the same thing but instead collapse the middle three objects to a point, the intermediate diagram becomes

$$ \require{AMScd} \begin{CD} & & C \times_D E \\ & & @VVV \\A \times_B C@>>> C \end{CD} $$

So, trying to perform this operation isn't local at all; it modifies the value of the diagram at the other two vertices.

It seems clear what the the abstract theory behind this sort of calculation should be; just factor the usual adjunction into a sequence of adjunctions.

But my interest in such things is very much not in the abstract — these are the sorts of operations one would like to have as a practical calculus of diagrams.

So my question is if such a calculus is known? Is there worked out how to predict and recognize which sorts of operations really should be local? Or for those operations that are not local, to easily work out how the rest of the diagram gets modified?

(and the bonus question: how much of this carries over to homotopy limits?)