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In this article we continue our investigation of the M\"obius-invariantWillmore flow, starting in arbitrary smooth and umbilic-free initial immersions$F_0$ which map some fixed compact torus into $\mathbb{R}^n$ respectively$\mathbb{S}^n$. We investigate the behavior of flow lines $\{F_t\}$ of theM\"obius-invariant Willmore flow in $\mathbb{S}^3$ starting with Willmoreenergy below $8\pi$, as the time approaches the maximal time of existence$T_{max}(F_0)$ of $\{F_t\}$. We particularly investigate the formation ofsingularities of flow lines of the M\"obius-invariant Willmore flow in bothfinite and infinite time. We will see that every limit surface of a flow lineof the MIWF can be identified with the support of a certain integral$2$-varifold $\mu$ in $\mathbb{R}^4$ and that the support of this varifold$\mu$ has either degenerated to a point or is homeomorphic to either a sphereor a compact torus in $\mathbb{S}^3$. In the non-degenerate case in which thelimit surface is a compact torus, it can be parametrized by a uniformlyconformal bi-Lipschitz homeomorphism of class $(W^{2,2}\capW^{1,\infty})(\Sigma)$. Under certain additional conditions on the consideredflow line, such a non-degenerate limit surface of $\{F_t\}$ can be parametrizedby a uniformly conformal diffeomorphism of class$W^{4,2}(\Sigma,\mathbb{R}^4)$. Finally, if the initial immersion $F_0$ of aflow line is assumed to parametrize a smooth Hopf-torus with Willmore energysmaller than $8\pi$, then we obtain stronger types of convergence of particularsubsequences of $\{F_t\}$ to uniformly conformal $W^{4,2}$-parametrizations ofcertain limit Hopf-tori, and this insight yields a simple criterion for full$C^m$-convergence of such a flow line of the MIWF to a smooth parametrizationof the Clifford-torus - up to a M\"obius-transformation of $\mathbb{S}^3$ - forevery fixed $m \in \mathbb{N}$.

Singularities and full convergence of the M\"obius-invariant Willmore flow in the $3$-sphere