Suspended particles flowing through complex porous spaces exhibit clogging mechanisms determined by factors including their size, deformability, and the geometry of the confinement. This study describes the clogging of rigid particles in a microfluidic device made up of parallel microchannels which taper from the inlet to the outlet, where the constriction width is approximately equal to the particle size. This geometry summarizes the dynamics of clogging in flow channels with constrictions that narrow over multiple length scales. Flow tests are conducted at constant driving pressures for different particle volume fractions, and a power-law decay which appears to be peculiar to the tapered geometry of the channels is observed in all cases. In comparison with non-tapered channels, the power-law exponent shows the flowrate decay rate is significantly lower in a tapered channel. Also, micrographs of the clogged channels reveal the clogs do not grow continuously from their initial points of inception. Rather, multiple clogs with increasing number of particles in their cross section are successively formed as the cake grows in each microchannel. Changes in particle volume fraction at a constant driving pressure affect the clogging rate without impacting the underlying clogging dynamics. Unexpectedly, analyses of the particles packing behavior in the microchannels and post-clogging permeability reveal the presence of two distinct regimes of driving pressure, though only a small portion of the total device volume and channels surface area were occupied by clogs, regardless of the particle volume fraction.