The Poynting vector is not well behaved with respect to special relativity.
It is not part of any four-vector.
Of course the same can be said of the E-vector and the B-vector; they
are not well behaved either.
The Poynting vector can be expressed as the spacelike rows of the timelike
column of a 4x4 matrix "T". The other row in that column is the Poynting
energy density. In particular, if we have observer with 4-velocity u, then
T(u) is 4-vector representing the energy density and energy-flow density.
The spacelike part of the stress-energy tensor is the plain old stress tensor.
This is the same sort of stress tensor that you encounter in the theory of
elasticity, or in fluid dynamics.
This electromagnetic stress is what you observed when you were 10 years old,
playing with a magnet and iron filings: There is tension along each field
line and repulsion between field lines.
The xy component of stress is the flow (in the x direction) of the y-component
of momentum (or vice versa).
Promoting the 3x3 stress tensor to a 4x4 stress-energy tensor gives a
relativistically correct picture of what's going on.