How does cross-border inter-operability work in relation to digital signatures?

Clearly, all certificates, not to mention technology applications, cannot and would not be issued by a single CA. Multiple CA's do and must exist. Inter operability between CAs- national and cross-border - has been addressed as Cross Certification. As per Information Technology (Certifying Authority) Rules, 2000 

The licensed CA shall have arrangement for cross certification with other licensed CAs within India, which shall be submitted to the Controller before the commencement of their operations as per rule 20. Disputes arising as a result of such arrangements shall be submitted to CCA, India for arbitration or resolution.

The arrangement for cross certification by the licensed CA with a foreign CA along with the application shall be submitted to CCA, India. The licensed CA shall not commence cross certification operations unless it has obtained the written or digital signature approval from CCA, India.