The Fourier-Jacobi-decomposition of Eisenstein series of Klingen type

The space of Siegel modular forms of degree $n$ and weight k has a
decomposition in a direct sum M_n^k=\oplus_{m=0}^{n}M_{n,m}^k, where the
space M_{n,m}^k corresponds to the space of cusp forms of degree m and
weight k. A Siegel  modular form of degree n has Fourier-Jacobi
expansions of degree r<=n. The spaces of Jacobi forms have (by work of
Dulinski) similar decompositions.

I want to describe how these decompositions fit together, meaning to
compute the decomposition of a Fourier-Jacobi-coefficient of a Siegel
modular form in M_{n,m}^k.