  A theoretical framework to investigate the time evolution of the quantum
entanglement due to the dynamical Lamb effect between $N$ superconducting
qubits coupled to a coplanar waveguide in the presence of different sources of
dissipation is developed. We quantitatively analyze the case of $N=2$ and $3$
qubits under the assumptions of single switching of the coupling and absence of
dissipation within a perturbative approach. The same systems are analyzed for
the general case of periodic switching of the coupling in the presence of
dissipation via numerical calculations. Different measures of entanglement
compatible with mixed states are adopted. It is demonstrated that the different
measures show different level of details of the latter. The concurrence and the
negativity are obtained in the two qubits case, the three-$\pi$ and the
negativity in the three qubits case. It is shown that time-dependent
Greenberger-Horne-Zeilinger states can be created even in presence of
dissipation. To maximize the quantum entanglement between the qubits, the
effects of tuning several parameters of the system are investigated.
