I know of no cause of action which would have a reasonable possibility of success.
Your wife has a cause of action against him for nonsupport, and if her dissolution decree and support order were prepared adequately, she should have a large judgment against him right now (or she could go to court to establish the amount of that judgment for back support, plus interest at 12%.) A judgment is all that you would be able to obtain. If you know how to collect a judgment, help your wife collect on hers, rather than pursuing fruitless litigation on your own.
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RCW 26.16.205 (Liability for family support — Support obligation of stepparent) provides: "The expenses of the family and the education of the children, including stepchildren, are chargeable upon the property of both spouses or both domestic partners, or either of them, and they may be sued jointly or separately. When a petition for dissolution of marriage or state registered domestic partnership or a petition for legal separation is filed, the court may, upon motion of the stepparent, terminate the obligation to support the stepchildren. The obligation to support stepchildren shall cease upon the entry of a decree of dissolution, decree of legal separation, or death."
So, WA laws require you to support stepchildren who are living with you.
You must have known your future wife has minor children when you married her. If you did not want to support the stepchildren, you should not have married her. If you do not want to support the stepchildren now, you need to dissolve your marriage with their mother.
Your wife likely has an order of child support that requires the children's father to pay her child support. She can use that order of child support to collect from the father.
Given that the father is out of the country, there may be no practical way to collect.
You and your wife should review the specific facts with your respective attorney to find out the legal options.
There is no basis in the law for such a suit -- you have no standing. for any kind of action to progress through the court, your wife would be the party of interest. You are not the children's parent and as such had no financial responsibility for them.
The information is for general information purposes only. Nothing stated above should be taken as legal advice for any individual case or situation.