It is well known that straight static cosmic strings don't produce any gravitational effects on test-particles, and that static flat domain walls are repulsive. This can be seen from the linearized theory of Einstein's equation, in the case of static weak fields: $$\tag{1} \square\, \bar{h}_{\mu \nu} = - \, \nabla^2 \, \bar{h}_{\mu \nu} = -\, 8 \pi G \, T_{\mu \nu}, $$ where $$\tag{2} \bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \, \eta_{\mu \nu} \, h, $$ and $\phi = h_{00}/2$ is the newtonian potential. Playing a bit with these, and the energy-momentum of a static anisotropic fluid, give the following: $$\tag{3} \nabla^2 \, \phi = 4 \pi G (\rho + 2 \sigma + p). $$ For a cosmic string, tangential pressure is negative: $p = -\, \rho$, and orthogonal pressure is $\sigma = 0$, so the RHS of (3) is 0 (the string doesn't have a gravitational effect). For a cosmic wall we have $\sigma = -\, \rho$ and $p = 0$ so (3) is negative (the wall is repulsive).
But then I wonder how true is the conclusion. What if the string or wall is moving, twisting and deforming (non-static)?
Is it always true (I don't think so) that any thin cosmic string doesn't attract/repulse? And is it true that any thin cosmic domain wall is repulsive? What is the general case?
EDIT: We need to distinguish two motion states for the strings and walls: static and dynamical (i.e. with arbitrary motions). If the strings/walls are curvy and twisted, but still static, the above equations appear to say that the conclusion is still true (i.e. strings don't attract or repel, while static curvy walls are still repulsive). For dynamical strings/walls, the conclusion appears to be false.