Wa alaikum salaam wa rahmatulahi wa barakatuh brother,
Masha Allah, you are approaching this issue in a respectable manner for the sake of Allah and to the best for your children. It is unfortunate that she went ahead and made the move without consulting you as you can have discussed matters relating to you providing financial support ahead of time along with some conditions you would prefer her to abide by when it comes to matters of the children. However, as this did not happen, you have to deal with the matter retrospectively.
Of course, you want what’s best for children, especially when it comes to their Deen and you are aware that you are still responsible for providing for them financially. However, with the fear of how they might be influenced by the West you are concerned that this financial provision will only encourage them to stay and be potentially influenced by factors in the West that are not compatible with Islam.
This is a big choice to make as whichever choice you make there may be some dissatisfaction in the light of Islam; either you are left not providing for them as you should, yet potentially protecting them from harmful exposure, but then you would be disregarding your duties as a father, but possibly supporting them in remaining somewhere that wont be good for them.
Beyond these obvious concerns, there are other things to think about that might make moving forward for you easier. As their father, you should have some say in the children’s upbringing, even if you are divorced and live apart from them. The extent to which you are obliged to be involved after a divorce is a matter that I would urge you to seek advice from a scholar who will be able to provide you with the correct rulings in light of the Qur’an and Sunnah so that you can rest assured that whatever you do will be that which is obligated upon you.
This will also enable you to provide a more convincing stance in approaching your ex wife. If it is a matter that she does not agree to, then you could seek assistance from an imam or scholar of knowledge to approach her either with you, or separately if necessary. From afar this is something that would obviously have to take place over the phone or Internet. Otherwise, if you are able to engage someone close to her, then they could approach her on the matter in person.
Do also be aware that there may other matters relating to the laws of the country of both the country you married in, as well as where she is now, and the manner in which you married (Islamic, or civil), both of yours and your children’s nationalities. These are issues that may lead to the need for legal involvement should it be deemed that you have the right to withdraw them. Islamically, the children’s gender and age will also be relevant in the amount of control you have. Again, this is something to seek advice from a learned scholar on.
If it is deemed that it is acceptable for her to stay with them in the West, do understand that whilst there are many aspects that are quite against the principles of Islam, there are also many positives also. There are good Muslim communities, good Muslim schools and plenty of opportunities to thrive as a Muslim in ways that are compatible with Islam. Even in Muslim countries their exists opportunities for Muslim to become court and go astray often in the light of cultural matters. This is where cooperation with your ex wife, working together for the sake of Allah and your children will be good for your peace of mind and their moral upbringing, even in your absence.
Make arrangements to talk to them often via video calls. This way you can consistently monitor their state and make sure that they are indeed staying strong in their Deen. If you notice anything concerning, then this is where you could talk to their mother and find out what you can do to improve things. This way, you can be as involved as possible in their lives from afar whilst also monitoring their progress.
As you are concerned that sending any financial support their way will only encourage and enable them to remain in the West perhaps, if is that it is deemed that they can and will stay, but that you must provide for them financially then you could consider controlling the way you support them. Rather than sending their mother money each month, instead, consider offering to pay for the things that they need.
For example, if they are attending an Islamic school and fees need to be paid, then you could offer to pay for them directly and pay straight to the school. Likewise, you could order items such as clothing online to be delivered directly to their address. This way, you can still support them financially but in a way that you know is going directly towards meeting their needs than funding other things that might not be compatible with Islam.
Do let the children and your ex wife know also that the door is always open should they wish to return anytime, or should it be deemed from scholarly advise that they should return to you. This way things still remain gentle and supportive and in line with the advice you receive between all parties without being over confrontational, especially with the wellbeing of children being at stake.
May Allah reward your efforts to raise your children well even after divorce. May He make them the coolness of your eyes in this life and the next.