I'm not really following your question well. It is well known that accelerating charges and non-stationary currents have a positive flux of Poynting vector over a closed surface around them. This is what we call electromagnetic radiation and it carries energy out of the system.
If you have static charges, they produce no magnetic field. So the Poynting vector is zero. If you have a steady current in an ideal wire, there is no electric field and the Poynting vector is zero.
If you consider an arbitrary box and the closed surface integral of the Poynting vector is positive, it means the electromagnetic fields are carrying energy out of the box. That does not necessarily make it "radiation".
For example if you look in the Wikipedia page on the Poynting vector you will see examples of influx and outflux of the Poynting vector in DC circuits (with resistance), which we wouldn't term radiation because the fields are stationary.