The present invention relates to a dispensing device comprising: a) a container comprising a deformable wall, and b) a composition stored in the container, and comprising, in a cosmetically acceptable medium: i) at least one aqueous phase, and ii) optionally at least one oily phase, iii) at least one structuring agent; the said composition having a stiffness modulus G*>5000 Pa, and c) a dispensing head closing off the container and comprising an application wall defining at least one product dispensing orifice which is a slit. The present invention also relates to a cosmetic process for treating and/or caring for human keratin materials, characterized in that it consists in applying to the surface of the keratin material a composition dispensed via the device as defined previously. The present invention also relates to a cosmetic process for treating human perspiration and/or perspiration-related body odour, which consists in applying to the surface of a human keratin material a composition dispensed via the device as defined previously.