All current MUMPS language implementations have the ability to specify MUMPS programs that respond to a TCP/IP connection. The native MUMPS database is structured as a hierarchy of ordered multi-key and value pairs, essentially a superset of the NoSQL paradigm.
KB/SQL is a group of programs that respond to SQL/ODBC queries, translate them into this MUMPS "global" data queries, retrieve and consolidate the results from MUMPS, and then send back the data in the form that the SQL/ODBC protocol expects.
If you have the permissions/security authorization for your implementation that allows you to create and run MUMPS programs (called "routines"), then you can respond to any protocol you desire from the those programs. MUMPS systems can produce text or binary results on a TCP/IP port, or a host operating system file. Many vendors explicitly keep you from doing this in their contracts to provide healthcare and financial solutions.
To my knowledge the LINQ syntax is a proprietary Microsoft product, although there are certainly LINQ-like Open Source efforts out there. I have not seen any formal definition of a line protocol for LINQ, but if there is one, a MUMPS routine can be written to communicate using that protocol. It would have to do something similar to KB/SQL however, since neither the LINQ syntax nor the SQL syntax are very close to the native MUMPS syntax.
The MUMPS data structuring and storage mechanism can be mechanically translated into the an XML syntax. This may still require an extensive effort as it is highly unlikely that the vendor of your system will provide a DTD defined for this mechanically created XML syntax, and you will still have to deal with encoded values and references which are stored in the MUMPS based system in their raw form.