The string equations of motion and constraints are solved for a ring-shaped ansatz in cosmological and black hole spacetimes. In FRW universes with arbitrary power behavior [$R\left(X^0\right)=a{\left|X^0\right|}^{\alpha }$], the asymptotic form of the solution is found for both $X^0\to 0$ and $X^0\to \infty$ and we plot the numerical solution for all times. Right after the big bang ($X^0=0\mathrm{}$), the string energy decreases as $R{\left(X^0\right)}^{-1\mathrm{}}$ and the string size grows as $R\left(X^0\right)$ for $0<\mathrm{}\alpha <1\mathrm{}$ and as $X^0$ for $\alpha <0\mathrm{}$ and $\alpha >1\mathrm{}$. Very soon [$X^0\sim 1$], the ring reaches its oscillatory regime with frequency equal to the winding and constant size and energy. This picture holds for all values of $\alpha$ including string vacua (for which, asymptotically, $\alpha =1\mathrm{}$). In addition, an exact nonoscillatory ring solution is found. For black hole spacetimes (Schwarzschild, Reissner-Nordström, and stringy), we solve for ring strings moving towards the center. Depending on their initial conditions (essentially the oscillation phase), they are absorbed or not by Schwarzschild black holes. The phenomenon of particle transmutation is explicitly observed (for rings not swallowed by the hole). An effective horizon is noticed for the rings. Exact and explicit ring solutions inside the horizon(s) are found. They may be interpreted as strings propagating between the different universes described by the full black hole manifold.

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