Some systems do not support custom fields that are User/Contact Picklist but they do support participants.

Use this script to copy custom field values/definitions into the Matter Participants table.

--We need to copy some records from custom fields into matter participants.
INSERT INTO __M_Matters_Participating_Entities (
    Id,
    Final_Matter_Id,
    Final_Participant_Type,
    Final_Participant_Id,
    Final_Role
)

SELECT
    CONCAT(V1.Id, V2.Id),
    V2.Final_Parent_Id,
    V1.Final_Kind,
    V2.Final_Value,
    V1.Final_Subject
FROM
    __M_CustomField_Definitions V1,
    __M_CustomField_Values V2
WHERE 1=1
    --Do the conversion for custom fields that are Contact or User picklists
    --If you only want Users, then adjust the following.
    AND V1.Final_Kind IN ('__M_Contacts', '__M_Users')
    --And extend the Matters table.
    AND V1.Final_Parent_Type IN ('__M_Matters')
    AND V2.Final_CustomFieldDefinition_Id = V1.Id