Is it permissible or forbidden for a husband to travel with his wife and young daughters to a non-Muslim country, not for permanent residency, but rather for the purpose of registering the daughters to obtain money, then returning after a few months to their Muslim country of residence, with the aim of buying a house for the daughters, covering their educational expenses, and starting a business project?

If the purpose of clocking in and then traveling is to obtain money from the employer or the state because your family is with you, then there is no harm in that. However, if the intention is to defraud the employer or the state by clocking in and then traveling without their knowledge, in order to obtain a bonus or similar, by claiming your presence when you are not there, then this is forbidden deception, and it is impermissible to take compensation for it.