  Regular black strings solutions associated to a dynamical
Friedmann-Robertson-Walker braneworld are obtained as a particular case of a
regular bulk metric, in the context of a variable brane tension. By analyzing
the 5D Kretschmann invariants, we show that the variable brane tension is
capable to remove bulk singularities, along some eras of the evolution of the
Universe. In particular, the black string time-dependent profile in the bulk is
analyzed in the context of the McVittie metric on an E\"otv\"os fluid
braneworld, wherein the fluid dynamical brane tension depends on the brane
temperature. The whole bulk is shown to be regular for a Universe dominated by
non-relativistic matter or by relativistic matter/radiation, as the
cosmological time elapses. In a particular case the associated black strings
are shown to have finite extension along the extra dimension, what prominently
modify the higher-dimensional physical singularities. For a Universe dominated
by matter or radiation, we show that the brane singularities are removed along
the extra dimension, as an effect of the variable brane tension, corresponding
also to the vanishing of the black string horizon. In this case there is a
regular solution in the bulk. We show further that no additional physical
singularity is introduced in the bulk when the Universe is dominated by a
cosmological constant.
