Dynamic behaviors of droplets in asymmetric unilateral Y-junctions are experimentally studied. Three breakup regimes are categorized, namely no breakup, breakup with gaps and breakup with obstruction, and regime diagrams are constructed by the capillary number and the normalized droplet length. Influences of the bifurcation angle and the viscosity ratio on the two critical conditions between neighboring regimes are revealed. In particular, necking processes in the obstructed breakup regime are discussed and they are divided into three stages according to the evolution of the thinning rate. At the two previous stages driven by the two-phase flow, the necking processes show distinct characteristics under the influences of four different parameters, including capillary number, droplet length, viscosity ratio, and bifurcation angle, and the corresponding reasons are provided in detail. By contrast, the final pinch-off stage is driven by two different mechanisms depending merely on the physical properties of liquid systems. The neck collapsing of the low-viscosity droplet is driven by the potential flow while that of the high-viscosity droplet is driven by the two-fluid Stokes flow. Quantitative relations between the minimum neck width and the remaining time are built in the two situations and are found to coincide with the theoretical predictions, respectively.