There is a long-standing halachic debate whether Eilat is considered a part of Eretz Yisrael or not. According to Rav Aviner it is, and he lists a number of reasons to support his decision, while according to Rav Elyashiv it is not. I think the most common psak is that it is not halachically Eretz Yisrael, and this has ramifications regarding travel to Eilat from within Eretz Yisrael's halachic borders, and with regard to 2nd day of yom tov (though some who say it is not eretz yisrael still say to only keep one day of yom tov in Eilat).
This issue also has ramifications regarding divorce.
A divorced couple in Netanya went to a beis din to discuss this exact issue. The ex-wife, who is remarried, wanted to move to Eilat with her new husband and her three children. She said that her ex-husband does not pay his child-support payments regularly, and the job offer she received in Eilat with a higher salary would help her pay her bills. The ex-husband counter-argued that a) Eilat is outside of Eretz Yisrael and b) Eilat is a city with questionable moral standards and many problems with immodesty and is therefore inappropriate for her to take their children to live there.
While acknowledging that Eilat is far away and it would be difficult for the father to go to visit his kids, they concluded that Eilat is a part of Eretz Yisrael. At the same time, one cannot ignore the reputation Eilat has among the people. However, one can find similar issues in many places around Israel, like Tel Aviv and Netanya, and especially all along the coastal region of Israel, including Netanya, where the mother actually lives.
The dayanim concluded that just like Netanya has problems with beaches and immodesty, yet the mother finds it possible to live there and raise her kids properly in a religious community to be Torah-observant, so too in Eilat as well she is able to claim that she is moving to a Torah-observant community and raise her children there.
In the end the decision was to not let her move to Eilat because of the difficulty it would create for the father to visit his kids. Despite Eilat being, according to these dayanim, part of Eretz Yisrael.
(source: Bechadrei)
I wonder if they would have prevented her from moving to Kiryat Shmona or Metulla for the same reason. It seems reasonable to not allow a parent to move the children too far away to be reasonable for the other parent to visit, but the ex-husband should be pressured to fulfill his financial obligations towards child support payments. If she is considering moving to Eilat because she cannot afford to raise her children, the lack of child-support payments must be pretty serious.
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