Hi Jacob,
Traffic = radio traffic = words transmitted or words to transmit..
That being said, I have never heard "with" traffic. Some standard uses of traffic:
Unit 2, switch to channel 7 for your traffic.
Unit 2, standby for traffic.
Unit 2, do you have traffic?
Unit 2, what traffic are you referring to?
Unit 2, we didn't copy, repeat your traffic.
Now that being said, I can think of a situation where your "with" quote would make sense:
Medic 31 is travelling back to his quarters and has reported AIV (available in vehicle).
A call is put out for Medic 31.
As Medic 31 listens to the call details he passes in front of a house that is showing light smoke from the eaves.
He acknowledges the traffic with the words "Medic 31, 10-4 responding with traffic" (your quote)(meaning he is responding but has more to say).
Logical continuation of the exchange:
10-4, Medic 31, go ahead with your traffic.
Medic 31, reporting a house fire at 1234 Oak Street.
10-4 Medic 31, will dispatch Fire Units to 1234 Oak Street.
All of the above is predicated on your "with traffic" being an EXACT quote. If it was actually "to traffic" instead, there is no long explanation necessary.
Also, I don't know where you heard this, but they may have a standard for wording that is not universal in nature.
Rob