Sooo--would you argue that military personnel are all absentee parents when sent overseas?
The reality is that this father never walked out on his children. He accepted a position in Israel, where both parents have citizenship. It was his hope and desire that the whole family unit would move there together. Both parents have roots, and citizenship, in Israel. Both parents have family there. My understanding is that there may have been some contention between the parents regarding this decision. However, after a time, mom sold the family house, and cars and arranged to ship all of the family belongings to Israel. They cosigned an extended lease on a condo there. The children were enrolled in school. And then mom bolted with the kids. Now those are pretty much the facts as they appear in the original divorce proceedings and custody discussion--which took place in an international court due to the two country issue. The court ruled that due to the short time the children had lived in Israel, and despite the indications that this was clearly intended as permanent home for them, that their ties to the US had greater weight. There was no termination of parental rights, however, and the parents were awarded joint legal custody. The father pays child support, has a right to regular visitation and a role in major decision-making.
Now--this does not mean that the children may not perceive his actions as abandonment. Recall that their ages at the time were something like 4, 5 and 9 (or so). Feelings do not always follow facts. Children may well feel that a parent did not fight hard enough to maintain a relationship, or to remain in a marriage. Meanwhile they are privy to the day-to-day impacts of the broken marriage on the parent with physical custody. I fully understand, from friends' divorces, that a parent in pain carries an incredibly heavy load to support children's respect for the absent parent, particularly when they are angry, hurt or unresolved. Consider the possibility that mom in this case did not want the marriage to end so much as she wanted it to continue but under her own conditions and in the US. I would suggest that this is a very common scenario.
However, now, after five years, and after review of the record of controlling behaviors on the part of the mom, and particularly the overt efforts to sabotage anything in the way of healing therapy for the children--coupled with the reality that the father has fought across two continents consistently during that time, including multiple different visitation arrangements (including supervised visitation, not to mention surrendering his passport when he visits), I think it is time to recognize that these kids are being actively harmed by the current situation. I don't feel a need to lay out blame, however, I would point out that despite mom's obvious capability of arranging a full and complete schedule of lessons (tennis, tae kwan do, guitar, violin) for these children in addition to school, she has never of her own volition made counseling for the children a priority. In addition, they seem to be locked together as a block--they all take guitar lessons together, they all take violin together--one set of lessons even includes mom. Given age and gender differences, not to mention the likelihood of individual differences in temperament, interests, development and the like, this looks more pathological than healthy--feeding into an inability of the children to see themselves as individuals separate from their mother and from one another. Mom seems to be very threatened by the notion of any kind of healing therapy for either the children, or herself. Because she is stuck she has placed an enormous emotional load on her children to see the parents as being on opposite sides of a battle in which they must choose one or the other. Again--not healthy. Personally I hope that she takes advantage of the time she has been offered under the current situation to get some personal support and help. Because her children need an emotionally healthy mother, as well as needing a connection to their father.