What you could do is to use a dynamic parameter inside your particle material, which e.g. adds its value to your panners’ speed. Then inside your particles add a parameter dynamic and use the one you create in your material. With settings: ‘float uniform’ between two values, e.g. [-1, 1]. Tick ‘Spawn time only’. Then your particles should have random panner speed added to them.
Why would you want to use ribbons? If you are going for an abstract fire you could, but i’d go for flipbooks, alpha cutout panning, or shader magic like this.
If my explanation was too cryptic, i’ll try to make some screenshots later on when i’m home.
BTW: particle random node only works with gpu particles.