When contractor of a contract employee changes, it amounts to end of employment of such employee. New contractor should issue new letter of appointment or terms of service. Has this process been followed when contractors were changed? If no, then theses are sham contracts.
Such change of contractors is namesake as there are no letters issued by contractor (if at all).
If these were sham contracts then you can put up case of abolition of contract employment & claim direct employment.You have to approach Assistant Labour Commissioner (Central) with written down facts. At the end you may mention that since the contracts were sham, the ALC (Central) should pass order of abolition of contract & direct the employer to take you on his muster rolls.
Yes, technically new contractor may not continue with your contract employment. It is job of the existing contractor to terminate your contract employment, if at all.