As this question has been asked as an integrity question, I'm going to assume there is no legal obligation to pay it, otherwise this would be a legal question:
If the employee has earned the bonus and worked well, but decided to leave, they should be paid (morally and karmically). However if the funds take time to arrive from the client, there is no obligation on the company to be out of pocket until they arrive, or if the bonus isn't paid. I would make arrangements to pay the ex-employee when the contract is completed and the bonus is paid to the company.
If the emails suggest that the company has previously been expected to assume the risk of the bonus not being paid by the client, but the employee has waived the agreement to be paid on time, I feel there is a stronger case for getting paid sooner.
Good luck