I also tried update_response with an array (id of the response and the new token) but the new token does not update.

The idea was to do the propagation in three steps.

1. get_response_id (using old token)
2. update_response (using ids from get_response_id and try to update token)
3. set_participant_properties with new token (query done with old token)

Same in GUI : not an issue ? Desired behavior in my opinion.

If it's the desire behavior, Is there a way using the remote control function to update the token of a response ?

Since it does not seem to be an issue a follow-up on this here: https://forums.limesurvey.org/forum/can-i-do-this-with-limesurvey/125048-update-response-token-using-remote-control-functions

My opinion : update_response must allow updating «token»
If it's the case : no issue.

So far I was not able to make it run maybe I forget something or I'm doing it wrong. Can somebody validate that it should be working ?

2. update_response (using ids from get_response_id and try to update token)

This seems to work ?
Can you tell us the instruction tested ?

3. set_participant_properties with new token (query done with old token)

Can you tell us the function tested ?

2 Update response
$aResponseModifyed = array('id' => $id_response, 'token' => $next_multi_token);
$updatedResponse = $lsJSONRPCClient->update_response($sSessionKey, $surveyID, $aResponseModifyed);

3. set_participant_properties with new token (query done with old token)
   $aTokenData = array('token' => $next_token);
   $aQuery = array('token' => $current_token);
   $updated_participant = $lsJSONRPCClient->set_participant_properties($sSessionKey, $surveyID, $aQuery, $aTokenData);

2 not working
3 working

OK :
Confirm the issue : https://github.com/LimeSurvey/LimeSurvey/blob/957bf9a0c18fffb0bfc724e4ba8617046ebbe179/application/helpers/remotecontrol/remotecontrol_handle.php#L2973

OK to rename the title to «Unable to update token value of response with remote control» ?

I don't see how to rename the title. I will suggest a PR. Can we agree that modifying only the token should not be dependant on whether or not we have the permission to update responses ? In my opinion token is not a response to a survey.

PR sent: https://github.com/LimeSurvey/LimeSurvey/pull/1981