  We study the scattering amplitudes of massive Kaluza-Klein (KK) states under
toroidal compactification. We present a shifting method to quantitatively
derive the scattering amplitudes of massive KK gauge bosons and KK gravitons
from the corresponding massless amplitudes in the noncompactified higher
dimensional theories. With these we construct the massive KK scattering
amplitudes by extending the double-copy relations of massless scattering
amplitudes within the field theory framework, including both the BCJ and CHY
methods, and build up their connections to the massive KK KLT relations. We
present the massive BCJ-type double-copy construction of the $N$-point KK gauge
boson/graviton scattering amplitudes, and as the applications we derive
explicitly the four-point KK scattering amplitudes as well as the five-point KK
scattering amplitudes. We further study the nonrelativistic limit of these
massive scattering amplitudes with the heavy external KK states and discuss the
impact of the compactified extra dimensions on the low energy gravitational
potential. Finally, we analyze the four-point and $N$-point mass spectral
conditions and newly propose a novel group theory approach to prove that only
the KK theories under toroidal compactification can satisfy these conditions
for directly realizing massive double-copy in the field theory framework.
