It is customary to do the reference check once the candidature of the candidate is found satisfactory and an offer letter is issued. Conducting a reference check before the interview itself is unheard of.
When your ex-employee left the company, you must have issued him with the service-cum-employment certificate. For attending the job interview, this certificate is sufficient. A reference check can be done if the candidate is selected.
As far as appreciation is concerned, it cannot be given because the candidate needed to appear for the job interview. Appreciation is for doing exceptional work. It has to be given immediately after the employee demonstrates exemplary work or renders outstanding performance. It cannot be given as an afterthought.
Finally, if the ex-employee's future company wants to reference check, then it is their duty to send the proper form for the reference check through the official channel. You need to just fill in the blanks.
You can communicate your position to the ex-employee accordingly.
From India, Bangalore