  We present a comprehensive study of the spin excitations - as measured by the
dynamical spin structure factor $S(q,\omega)$ - of the so-called block-magnetic
state of low-dimensional orbital-selective Mott insulators. We realize this
state via both a multi-orbital Hubbard model and a generalized Kondo-Heisenberg
Hamiltonian. Due to various competing energy scales present in the models, the
system develops periodic ferromagnetic islands of various shapes and sizes,
which are antiferromagnetically coupled. The 2$\times$2 particular case was
already found experimentally in the ladder material BaFe$_2$Se$_3$ that becomes
superconducting under pressure. Here we discuss the electronic density as well
as Hubbard and Hund coupling dependence of $S(q,\omega)$ using density matrix
renormalization group method. Several interesting features were identified: (1)
An acoustic (dispersive spin-wave) mode develops. (2) The spin-wave bandwidth
establishes a new energy scale that is strongly dependent on the size of the
magnetic island and becomes abnormally small for large clusters. (3) Optical
(dispersionless spin excitation) modes are present for all block states studied
here. In addition, a variety of phenomenological spin Hamiltonians have been
investigated but none matches entirely our results that were obtained primarily
at intermediate Hubbard $U$ strengths. Our comprehensive analysis provides
theoretical guidance and motivation to crystal growers to search for
appropriate candidate materials to realize the block states, and to neutron
scattering experimentalists to confirm the exotic dynamical magnetic properties
unveiled here, with a rich mixture of acoustic and optical features.
