  Our main interest is the evolution of domain walls of the Higgs field in the
early Universe. The aim of this paper is to understand how dynamics of Higgs
domain walls could be influenced by yet unknown interactions from beyond the
Standard Model. We assume that the Standard Model is valid up to certain, high,
energy scale $\Lambda$ and use the framework of the effective field theory to
describe physics below that scale. Performing numerical simulations with
different values of the scale $\Lambda$ we are able to extend our previous
analysis and determine its range of validity.
  We study domain walls interpolating between the physical electroweak vacuum
and the vacuum appearing at very high field strengths. These domain walls could
be formed from non-homogeneous configurations of the Higgs field produced by
quantum fluctuations during inflation or thermal fluctuations during reheating.
  Our numerical simulations show that evolution of Higgs domain walls is rather
insensitive to interactions beyond the Standard Model as long as masses of new
particles are grater than $10^{12}\ \textrm{GeV}$. For lower values of
$\Lambda$ the RG improved effective potential is strongly modified at field
strengths crucial to the evolution of domain walls. For instance its minima
become degenerate for $\Lambda$ around $10^{11}\ \textrm{GeV}$. We find that
even in the case when the minima of the potential are nearly degenerate Higgs
domain walls decayed shortly after their formation for generic initial
conditions. On the other hand, in simulations with specifically chosen initial
conditions Higgs domain walls can live longer and enter the scaling regime.
  We also determine the energy spectrum of gravitational waves produced by
decaying domain walls of the Higgs field. For generic initial field
configurations the amplitude of the signal is too small to be observed in
present and planned detectors.
