In this paper, we are concerned with the large N limit of linear combinations of entries of Brownian motions on the group of N by N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time are concerned, giving rise to various limit processes, in relation to the geometric construction of the unitary Brownian motion. As an application, we recover certain results about linear combinations of the entries of Haar distributed random unitary matrices.