  We consider a quite general problem concerning a linear free oscillation of a
discrete mass-spring-damper system. This discrete sub-system is embedded into a
one-dimensional continuum medium described by the linear telegraph equation. In
a particular case, the discrete sub-system can move along the continuum one at
a sub-critical speed. Provided that the dissipation in both discrete and
continuum sub-systems is absent, if parameters of the sub-systems are
constants, under certain conditions (the localization conditions), a
non-vanishing oscillation localized near the discrete sub-system can be
possible. In the paper we assume that the dissipation in the damper and the
medium is small, and all discrete-continuum system parameters are slowly
varying functions in time and in space (when applicable), such that the
localization condition is fulfilled for the instantaneous values of the
parameters in a certain neighbourhood of the discrete sub-system position. This
general statement can describe a number of mechanical systems of various
nature. We derive the expression for the leading-order term of a universal
asymptotics, which describes a localized oscillation of the discrete
sub-system. In the non-dissipative case, the leading-order term of the
expansion for the amplitude is found in the form of an algebraic expression,
which involves the instantaneous values of the system parameters. In the
dissipative case, the leading-order term for the amplitude, generally, is found
in quadratures in the form of a functional, which depends on the history of the
system parameters, though in some exceptional cases the result can be obtained
as a function of time and the instantaneous limiting values of the system
parameters. Finally, we have justified the universal asymptotics by numerical
calculations for some particular cases.
