The constraints in your assembly probably don't do quite what you intend. If you view each imported part group in the browser window, then you'll see that they each have one DOF remaining. Since they're constrained at only two points, you can rotate them about the line between those points, dragging them out of plane (e.g. by Shift+dragging a point on the part). They of course are also all in one plane, and therefore interfering.

One possible way to constrain each imported part in your model is as follows, starting from the usual 6 degrees of freedom:

(a) Constrain the part to lie parallel to your reference xy plane, by constraining the part's z axis parallel to our z axis. The part now has 4 DOF: rotation and translation within the xy plane, and translation normal to the xy plane.

(b) Constrain point on plane, point on plane face, or point-plane distance to put the part at the desired z, to stop the links from interfering like in your model. The part now has 3 DOF, rotation and translation in the xy plane only.

(c) Switch from constraining in 3d (the default, or Sketch -> Anywhere in 3d) to constraining projected into the xy plane (select the xy plane, then Sketch -> In Workplane).

(d) Constrain the part onto the skeleton of the linkage, for example with a point-coincident constraint for translation, and a point-on-line or parallel constraint for rotation. The part is now fully constrained.

Of course, other sets of constraints would work too. This is probably the easiest set for me to understand, though. Whatever constraints you use, I'd suggest always looking in the browser window to confirm that the parts end up fully constrained (0 DOF).

Let me know if that works for you, or if you have any other questions.

You skipped step (c) above. Those two points are in different planes, so there's no way to constrain them truly coincident. You can constrain them coincident projected into the xy plane, though, and that will do what you want.

By default, sketch-in-workplane groups have constraints projected into that workplane, and other groups (like that import / assemble group) have constraints in 3d. You can always specify the behavior that you want, though, as described at

http://solvespace.com/constraints.pl

around "So far, we have...".