Midi programe change

You wrote:
The problem I have is that the midi data from musescore changers the selected sound in the synth for another.

OK... so MuseScore is primarily a musical score editor that produces sheet music. It does have playback capabilities which can be considered as a type of "audio checker" - analagous to the "spell checker" of a text editor.
Regarding internal playback, MuseScore's built-in synthesizer, together with its native soundfont, is GM (General MIDI) compliant. That means if you, for instance, need help and attach a MuseScore (.mscz) file to a forum post, a fellow MuseScorer can open it, play it in MuseScore, and hear what you hear. So... if your file is of a solo piano, that's what everyone else hears.

Now... when you say you have "midi data" from MuseScore, presumably that means you have exported your score to MIDI format.
You then open this MIDI file - e.g., of a piano solo - in Reaper and what? You hear a different (brand of / type of) piano? A saxophone?
Some non-GM sound (e.g., yodel) available only in Reaper?

If you play the MuseScore exported MIDI in a basic media player (VLC, MPC-HC), does it not sound correct?

I use Reaper but not Carla and Jack. So I don't know if this will help or make sense. ( I noticed that Carla has a number of midi controls including a return of midi data that may be at fault) I am not sure whether you are streaming midi or audio out, however, because you "hope to have Msc play the sounds in Reaper" could mean either. So, i will assume that midi comes from Msc and the audio is produced in Reaper. If so:

The problem could be dealt with on the Reaper end. Any number of vst's "snapback" to their original presets upon stop or play. This can be tackled in reaper preferences as best as possible by experimenting in audio playback preferences with stop/playback options or note on/off overides or manually setting a cc parameter to -1 to halt resets. You can try deselecting "control-enabled" in the midi-in device.