The motion of flocculated fibres in a streaming suspension is governed by the balance of the network strength and hydrodynamic forces. With increasing flow rate through a channel, (1) the network initially occupying all space, (2) is then compressed to the centre, and (3) ultimately dispersed. This classical view neglects fibres-fines: we find that the distribution of these small particles differs in streaming suspensions. While it is known that fibre-fines can escape the fibre network, we find that the distribution of fibre-fines is non-homogenous in the network during compression: fibre-fines can be caged and retarded in the streaming fibre network. Hence, the amount of fibre-fines is reduced outside of a fibre network and enriched at the network’s interface. Aiming on selectively removing fibre-fines from a streaming network by suction, we identify a reduction of the fines removal rate. That documents a hindered mobility of fibre-fines when moving through the network of fibres. Additionally, we found evidence, that the mobility of fibre-fines is dependent on the fibre-fines quality, and is higher for fibrillar fines. Consequently, we suggest that the quality of fibre-fines removed from the suspension can be controlled with the flow regime in the channel. Finally, we present a phenomenological model to compute the length dependent fibre distribution in an arbitary geometry. For a fibre suspension channel flow we are able to predict a length-dependent fibre segregation near the channel’s centre. The erosion of a plug of long fibres was however underestimated by our model. Interestingly, our model with parameters fitted to streaming fibre suspension qualitatively agreed with the motion of micro-fibrillated cellulose. This gives hope that devices for handling flocculated fibre suspensions can be designed in the future with greater confidence.