We show that concurrent atomistic/continuum displacement-coupling methods are inherently unstable in the context of fully dynamic problems at finite temperature. The fundamental origin of the instability is the displacement coupling of two material descriptions of which one exhibits a traction-compression asymmetry. Since this asymmetry applies to all realistic interatomic potentials, the results and conclusions should apply to all displacement-coupling multiscale methods. A few approaches to mitigate or completely avoid the instability are discussed including a promising new method for coupling local and non-local material descriptions in a ghost force-free, dynamic and stable way that does not require a well-defined Hamiltonian of the system. (C) 2015 Elsevier Ltd. All rights reserved.