For your information:
I just got back from my Home Country (by birth, not citizenship).
POE officer at JFK looked at my RT passport and asked what was a purpose of my travel. I answered. Then he noticed that a country code on the first page of my RTD matches my tourist visa asked with dazzled look on his face: “y o u a r e a n a s s y l e e?”
I said: “yes”
“ ….and you went back on a vacation to your home country ???” I said: “ I was born there, but I am a citizen of XX country” He nodded and typed something in the computer. Then he looked at my children’s passports, stamped them, returned to me and asked again: “so, you are an assylee?”. “Yes, sir, am I an assylee, a derivative assylee.”
“OK”
“ Is it your first time traveling with RTD?” I said it was my second.
More typing, then he asked once again what country citizen am I, I repeated my answer. Then he asked me to follow him (to the other room) and take a seat. I waited there for about 30 minutes. Without any more questions my RTD was stamped and the new I-94 attached.
Just in case I had a copy of my old expired national passport (proof of the citizenship) and approved I-730 notice (proof of derivative status), but nobody asked.