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We study representations of lattices of $\mathrm{PU}(m,1)$ into $\mathrm{PU}(n,1)$. We show that if a representation is reductive and if $m$ is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic $m$-space to complex hyperbolic $n$-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into $\mathrm{PU}(n,1)$ of non-uniform lattices in $\mathrm{PU}(1,1)$, and more generally of fundamental groups of orientable...