If the 'new' equipment is different the fittings are NOT compatible. My current supplies cannot be used unless I come across a very special adapter fitting. I have heard that some Clinics have access to such a fitting to use for transient patients. I haven't heard if it is available for patient use.
I assume you are talking about PD. The Fresenis and Baxter connections to the PD gut tube are different. When I was on PD, they gave me a disposable adapter and case to take with me if I ever was hospitalized in case the hospital was using the Baxer system.
As to switching clinics - You would keep supplies you already have (they are treated as potentially contaminated once in a patient house). Unless you were with the same company, you would almost certainly return your machine and get one from the new clinic. You would have to convince the MD at the new clinic to write orders for home treatment, and I expect the home training RN would want to see you do a treatment to make sure (s)he felt comfortable with your skills.
I know here in the Democratic Peoples Republic of Massachusetts all dialysis clinics must either have a home treatment program, or an affiliation with a clinic that offers home service. See 105 CMR 145.320 and 105 CMR 145.800 - 105 CMR 145.850 for details. Unfortunately, the CMR is not specific regarding modality, so a clinic might be able to meet the requirement by having a PD program.