We study the transport of Brownian particles under a constant driving force and moving in channels that present a varying centerline but have constant aperture width (serpentine channels). We investigate two types of channels, solid channels, in which the particles are geometrically confined between solid walls and soft channels, in which the particles are confined by the potential energy landscape. We consider the limit of narrow, slowly varying channels, i.e., when the aperture and the variation in the position of the centerline are small compared to the length of a unit cell in the channel (wavelength). We use the method of asymptotic expansions to determine both the average velocity (or mobility) and the effective dispersion coefficient of the particles. We show that both solid and soft-channels have the same effects on the transport properties up to leading order correction. Including the next order correction, we obtain that the mobility in a solid-channel is smaller than that in a soft-channel. However, we discuss an alternative definition of the effective width of a soft channel that leads to equal mobilities up to second order terms. Interestingly, in both cases, the corrections to the mobility of the particles are independent of the Péclet number, and the Einstein-Smoluchowski relation is satisfied.