There are a couple of different potential outcomes, and there are examples of pretty much all of them in some places. If the ridge is roughly parallel to the subduction zone:

1. Option 1 is that the ridge doesn't actually subduct because subduction stops before the ridge gets there. Effectively the idea is that subduction is driven by the negative buoyancy of the subducted slab, which is a function of the age/temperature of the slab. The piece of lithosphere adjacent to an active ridge is pretty warm, young, and positively buoyant so it will resist subducting. Depending on the relative competition of forces what may happen is that subduction slows down as this young lithosphere approaches the ridge (resisting subduction) and then the slab rips off (i.e., it detaches) because the slab pull force overcomes the strength of the slab nearer the surface. This can effectively terminate subduction (no slab pull = no subduction). As to what happens from there, it will depend on the specific forces, but most likely the ridge might die and there will be a general reorganization. That reorganization might see a wholly different set of plate boundary kinematics or the subduction zone might "jump", keeping effectively similar broad scale kinematics but with the subduction zone in a different place. It might also jump and reverse polarity.
2. Option 2 is the ridge subducts and the slab detaches because there's nothing really connecting the other side of the ridge to the slab. The end result of this proceeds largely the same as above.

In terms of these geometries, the basic assumption was effectively option 2, but in detail, it's actually hard to get a ridge to subduct and option 1 is more favorable (e.g., Burkett & Billen, 2009). Semi-parallel ridge subduction does happen though, and for it to happen, usually some amount of complicated geometries and "3D effects" are required (e.g., Burkett & Billen, 2010).

If instead the ridge is very oblique or orthogonal to the subduction zone, the ridge will subduct and in many cases a "slab window" will open along the subducted segment of the ridge. You can picture the ridge effectively unzippering down the length of the subduction zone, kind of like this. This makes some specific predictions about what you would see in the upper plate, specifically a gap in normal arc volcanism and instead magmatism that is more indicative of direct mantle interaction with the upper plate rocks.

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