Does the sponsor have the right, Islamically, to refuse the transfer of sponsorship in light of changing market conditions and wages, and to threaten to terminate the residency, especially given that he previously increased the salary? And is this considered exploitation?

The relationship between the employer and the worker is a contract of hire (ijara) based on mutual consent. It is binding on both parties until the end of the agreed-upon term, and neither party is permitted to unilaterally terminate it.

If the employer terminates the contract, the full wages for the entire period become due to the worker. If the worker terminates the contract while capable of working, they are not entitled to any compensation for the work they have done.

Therefore, the employer must fulfill the contract until its term expires. After the term expires, if the employer wishes to transfer the worker's sponsorship (kafaala), it is commendable. If they wish to renew the contract, they should agree upon a specific wage. It is not a condition that the worker be fully convinced by the wage; what is important is mutual consent to it, without coercion.