We provide a "toolkit" of basic lemmas for the
comparison of homotopy types of (homotopy) limits of
diagrams of spaces over finite partially ordered sets,
among them several new ones. In the setting of this
paper, we obtain simple inductive proofs that provide
explicit homotopy equivalences. (In an appendix we
provide the link to the general setting of diagrams of
spaces over an arbitrary small category.) We show how
this toolkit of old and new diagram lemmas can be used
on quite different fields of applications. In this
paper we demonstrate this with respect to -- the
"generalized homotopy-complementation formula" by
Björner~ Bjo89-1}, -- the topology of toric
varieties (which turn out to be homeomorphic to
homotopy limits, and for which the homotopy limit
construction provides a suitable spectral sequence),
-- in the study of homotopy types of arrangements of
subspaces, where we establish a new, general
combinatorial formula for the homotopy types of
"Grassmannian" arrangements, and -- in the analysis
of homotopy types of subgroup complexes.
