Encapsulated droplet formation resulting from the rupture and breakup of a compound liquid thread has numerous applications in industry. Understanding the process through which a compound thread will breakup and form droplets of a desired size is thus of great relevance and may help in developing refined experimental techniques. In this paper, we investigate the breakup and the instability of a viscous compound thread falling under gravity. The governing equations are formulated and a reduced one-dimensional set of equations is obtained using an asymptotic approach. The steady-state solutions, which are dependent on the axial distance along the thread, are determined using a modified Newton's method. Thereafter, we present a linear temporal instability analysis to study the effects of how key parameters affect the structure and onset of rupture. Finally, we investigate the nonlinear behaviour of waves along the free surface through using a finite difference scheme based on the Lax–Wendroff method and we obtain the breakup times and the sizes of main and satellite droplets.