  We prove weak laws of large numbers and central limit theorems of Lindeberg
type for empirical centres of mass (empirical Fr\'echet means) of independent
non-identically distributed random variables taking values in Riemannian
manifolds. In order to prove these theorems we describe and prove a simple kind
of Lindeberg-Feller central approximation theorem for vector-valued random
variables, which may be of independent interest and is therefore the subject of
a self-contained section. This vector-valued result allows us to clarify the
number of conditions required for the central limit theorem for empirical
Fr\'echet means, while extending its scope.
