What is Bench Warrant
Typically, judges issue bench warrants for persons deemed to be in contempt of court—possibly as a result of that person's failure to appear at the appointed time and date for a mandated court appearance. Bench warrants are issued in either criminal or civil court proceedings.
https://en.wikipedia.org/wiki/Arrest_warrant#Bench_warrant
How would Bench Warrant affect entering US.
the traveler will be arrested on the spot and transported in handcuffs to the locality where the bench warrant was issued.
