Be it known that HP ALM is only officially supported on Oracle and Microsoft SQL databases and currently has no support for MYSQL (and at this time no planned future support that we are aware of, as HP doesn't provide their roadmap for future releases and functionality/support). Please see the supported environments for current versions here: https://hpln.hp.com/page/alm-qc-enterprise-technical-specifications as usage of an unsupported environment can't guarantee correct functionality and/or operation/performance. Should issues arise in the future, support through official channels can be denied (it is at the support providers discretion) due to the use of a not ''officially'' supported environment, even though it appears to work and function as normal the use of an unsupported environment can't guarantee full functionality and compatibility as it is not fully tested and verified for correct functionality.
Assuming the environments are like in their design for layout of sufficient overhead and performance resources available, the performance should remain the same when moving environments, but such things as VPN or gateway usages can/may hinder performance as well as the usual traffic encountered through distance connectivity (assuming the servers are housed remotely and not locally as is the case with your current hosting situation) not to mention that the database they are utilizing isn't tested or guaranteed for performance functionality with HP ALM.
I would assume you should have no issue utilizing the .qcp movement of the projects (assuming they are within the useable size constraints and limits of the .qcp file), as that is an agnostic version (again this hasn't been tested on the MYSQL platform), but the usual methods of moving the projects may cause issues should the .qcp not be a viable method as moving between various DB versions using backups is an intricate and evasive operation (and once again should issues arise or be encountered, support can be denied based upon it being an unsupported environment for HP ALM).
Hope this helps, but I would recommend that they utilize a full officially supported environment to maintain guaranteed compatibility and future support should it be needed.