Is it permissible, in Islamic law, for a manager to refuse to grant an employee a waiver to transfer to another job, or to renege on promises of promotion and salary increase? And was sponsorship (kafalah) instituted to prevent an employee from earning his livelihood?

It is not permissible for the employer to break a promise. The questioner may continue his work while demanding a salary increase or non-renewal of the contract. The company has the right to prevent transfer if it is in accordance with the agreed-upon terms. One can complain to the responsible authorities to lift the injustice. Breaking promises and lying are characteristics of a hypocrite.