  In this work we revisit the existence, stability and dynamics of unstable
traveling solitary waves in the context of lattice dynamical systems. We
consider a nonlinear lattice of an $\alpha$-Fermi-Pasta-Ulam type with the
additional feature of all-to-all harmonic long-range interactions whose
strength decays exponentially with distance. The competition between the
nonlinear nearest-neighbor terms and the longer-range linear ones yields two
parameter regimes where the dependence of the energy $H$ of the traveling waves
on their velocity $c$ is non-monotonic and multivalued, respectively. We
examine both cases, and identify the exact (up to a prescribed numerical
tolerance) traveling waves. To investigate the stability of the obtained
solutions, we compute their Floquet multipliers, thinking of the traveling wave
problem as a periodic one modulo shifts. We show that in the general case when
the relationship between $H$ and $c$ is not single-valued, the sufficient but
not necessary criterion for stability change is $H'(s)=0$, where $s$ is the
parameter along the energy-velocity curve. Perturbing the unstable solutions
along the corresponding eigenvectors, we identify two different scenarios of
the dynamics of their transition to stable branches. In the first one, the
perturbed wave slows down after expelling a dispersive wave. The second
scenario involves an increase in the velocity of the perturbed wave accompanied
by the formation of a slower small-amplitude traveling solitary wave.
