The transient dynamics of a growing droplet in a yarn is explored following the spatiotemporal evolution of the three-phase contact line as well as the liquid-air interface with the help of videographic techniques and subsequent image analyses. The spontaneous capillary flow of liquids in a porous network is used to generate a droplet on the freely hanging end of a yarn whose other end is dipped continuously in a liquid reservoir. The growing droplet initially moves upward due to surface tension until the attainment of a critical volume, beyond which gravity is able to pull it downward until detachment. Based upon the spatiotemporal trajectory of the three-phase contact line of the droplet, the entirety of the associated growth dynamics can be divided in three distinct regimes, namely, "radial growth," "axial growth," and "motion" stages. The transition from one to the other is governed by the subtle interplay between the capillary and the gravity forces. Several experimental fluids are considered to elucidate the effect of the fluid properties on the transient contact line and interfacial dynamics of drops. The kinetics of the three-phase contact line and the radius of the droplet is found to follow two distinct exponential scaling laws, developed through the combination of the relevant forces. A mathematical model has also been proposed to predict the critical volume of the growing droplet in relation to its final volume, beyond which gravity controls the transient dynamics.