  We develop the number-conserving approach that has previously been used in a
single component Bose-Einstein condensed dilute atomic gas, to describe
consistent coupled condensate and noncondensate number dynamics, to an
$n$-component condensate. The resulting system of equations comprises, for each
component, of a generalised Gross-Pitaevskii equation coupled to modified
Bogoliubov-de Gennes equations. Lower-order approximations yield general
formulations for multi-component Gross-Pitaevskii equations, and systems of
multi-component Gross-Pitaevskii equations coupled to multi-component modified
number-conserving Bogoliubov-de Gennes equations. The analysis is left general,
such that, in the $n$-component condensate, there may or may not be mutually
coherent components. An expansion in powers of the ratio of noncondensate to
condensate particle numbers for each coherent set is used to derive the
self-consistent, second-order, dynamical equations of motion. The advantage of
the analysis developed in this article is in its applications to dynamical
instabilities that appear when two (or more) components are in conflict and
where a significant noncondensed fraction of atoms is expected to appear.
