  Motivated by the homometallic intertwining double-chain ferrimagnets
A_3_Cu_3_(PO_4_)_4_ (A=Ca,Sr,Pb), we investigate the low-energy structure of
their model Hamiltonian H=\sum_n_[J_1_(S_{n :1}_+S_{n :3}_)
+J_2_(S_{n+1:1}+S_{n-1:3}_)]\cdotS_{n:2}_, where S_{n:l}_ stands for the
Cu^{2+}^ ion spin labeled l in the nth trimer unit, with particular emphasis on
the range of bond alternation 0<J_2/J_1<1. Although the spin-wave theory,
whether up to O(S^1^) or up to O(S^0^), claims that there exists a flat band in
the excitation spectrum regardless of bond alternation, a perturbational
treatment as well as the exact diagonalization of the Hamiltonian reveals its
weak but nonvanishing momentum dispersion unless J_2_=J_1_ or J_2_=0. Quantum
Monte Carlo calculations of the static structure factor further convince us of
the low-lying excitation mechanism, elucidating similarities and differences
between the present system and alternating-spin linear-chain ferrimagnets.
