The directed, sodium-independent, movement of leukotrienes into, out of or within a cell, or between cells, by means of some agent such as a transporter or pore. Leukotrienes are linear C20 endogenous metabolites of arachidonic acid (icosa-5,8,11,14-tetraenoic acid) containing a terminal carboxy function and four or more double bonds (three or more of which are conjugated) as well as other functional groups.