The Navaid Remote Tuning will tune into any Navaid that is within range of the aircraft; then the Navaid will remain tuned until it is out of range of the aircraft, at which point the aircraft will start to search for new navaids that are within range.
Having a quick look at this in the sim, the ABA VOR has a range of approx. 190nm, so if the aircraft is out of range of any other navaids, the aircraft should automatically tune into ABA VOR when it is within 190nm. It did automatically tune for me, but I only did a quick local flight rather than the same flight as you mentioned.
I've wanted to do some Caribbean flying in the F70 for a while, so I'll add that flight to my list and keep an eye out for any oddities with the Navaid Remote Tuning!
Mark - Just Flight