There are a lot of things that you can do, many of which - as you are aware - cost a considerable amount of money. If you want a custody evaluation to be completed, it would be wise for you to immediately request that it be done, including offering to pay for it if you can find the money to do so. Additionally, if you are concerned about the financial claims, doing discovery (in particular document production) would advisable to obtain copies of your ex's bank statements. In the alternative, bank records could potentially be subpoenaed.
Depending on where the international move is, you might convince the judge to deny the international move. Where is the proposed move?
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You need to think seriously about hiring an international attorney. Best
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Custody/move-away cases are always the most expensive and emotionally draining in family law. The problem with such cases is that both parties present two extreme set of facts to the court, e.g. mom will claim father doesn't care about the child which is why he has spent "miniscule" time with the child; while, father will claim he has spent little to no time with the child because of mother's "alienating" behavior. In such cases, asking for a 730 custody evalution is justified. As my colleague correctly stated, such evaluations are expensive. If you feel that you can afford such an evaluation, then the next step is to ask the court for one--either as a separate motion or at your next hearing (or status hearing).
Remember, the basis of your request and/or the basis of the 730 expert's recommendations are going to be based on the "best interests of the child" legal standard.
If you cannot afford to hire attorney, please note that some attorneys provide coaching sessions. This is especially helpful in situations where a party needs legal assistance, but can't afford to hire an attorney to fully represent them. As you most likely already know, legal fees in such cases can be astronomical.
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