If the diagrams $X_\bullet$ and $Y_\bullet$ are constructed canonically, but not necessarily naturally, from spaces $X,Y$ where $X \simeq Y$, one technique to construct such a zigzag is pick $f: X \simeq Y$ and construct a diagram $\mathrm{cyl}(f)_\bullet$ based off the construction applied to the mapping cylinder $\mathrm{cyl}(f)$. If the construction of the diagram is natural with respect to inclusions, then we will have a zigzag $$X_\bullet \rightarrow \mathrm{cyl}(f)_\bullet \leftarrow Y_\bullet$$

which will often be a zigzag of equivalences. Similarly, if one was doing work in surgery theory or Waldhausen A-theory or the ilk, you might replace the role of the mapping cylinder with an H-cobordism.