Is there any obligation on employers to increase the holiday entitlement of TUPE-transferred staff to match that of their existing workforce?

No. When a TUPE-transfer takes place, the transferee inherits the contracts of employment of the employees employed by the transferor immediately before the transfer took place. This means that the employees have the right to transfer on the same terms and conditions of employment that they previously enjoyed with the transferor. The transferee also inherits the transferor's rights, powers, duties and liabilities (other than any criminal liabilities) under, or in connection with, those contracts of employment. Special rules apply to the transfer of occupational pension rights. However, TUPE does not grant employees contractual rights that are additional to the rights in place before the transfer. Thus, transferring employees have no right to an increase in holiday entitlement to match the holiday entitlement of the transferee's existing workforce.