Great question! How does oceanic lithosphere end up on the continents? Oceanic lithosphere is occasionally exposed in mountain belts in units call ophiolites. Ophiolites are sections of oceanic crust and lithospheric mantle that have been obducted during a collisional plate tectonic event.

There are several models for how ophiolite subduction works, but the basis of each comes down to density differences between the oceanic lithosphere and the underlying asthenosphere. subduction shows a very simplified cross section of a subduction zone. Oceanic lithosphere is subducting into the asthenosphere because it has become denser than the underlying asthenosphere (~3300 kg/m³) through cooling and thickening (as asthenospheric mantle turns into lithospheric mantle at the base of the plate). A mobile terrane (e.g., Japan), ocean island (e.g., Hawaii), or a seamount on the subducting plate may plug of the subduction zone. Because the column of seamount + oceanic lithosphere is less dense than the asthenosphere, it may float and be thrust up or obducted onto the margin and preserved as an ophiolite.

The other figure ophiolite (modified by Keary et al., 2009 from by Wakabayashi & Dilek, 2000) shows many more details of one of the competing models or ophiolite subduction.

I deviate from your question here, but I should mention that ophiolites are incredibly useful to geologists. They allow us to directly study the structure of oceanic crust and lithospheric mantle. They also provide us samples of mantle peridotite, which are relatively rare on the surface of the earth (also occasionally brought to the surface as xenoliths in mantle-derived volcanics).

Keary, P., Klepeis, K.A., Vine, F.J. (2009). Global Tectonics. Oxford: Wiley-Blackwell.
Wakabayashi, J. & Dilek, Y. (2000) Spatial and temporal relationships between ophiolites and their metamorphic soles: a test of models of forearc ophiolite genesis. In Dilek, Y. et al. (eds) Ophiolites and Oceanic Crust: new insights from field studies and ocean drilling program. Geo. Soc. Am. Sp Paper 349, 53-64.

This happens at locations where two plates come together to form a subduction zone. The upper plate consists of older oceanic lithosphere which can include sediments, mafic oceanic crust, and plutonic/-volcanic arcs. So when these two plates come together the stuff sitting on the down going slab can get scrapped off and emplaced by tectonic forces into the crust of the upper lithospheric plate. The rock bodies found preserved this way are almost always in fault contact. Serpentinite, a ductile rock is >90% the mineral serpentine a weak sheet silicate with a good cleavage as in mica...it can peel apart easier than other crustal minerals and because it contains H₂O in the form of hydroxyl (OH) it is soft and ductile. Serpine forms from the oxidation and hydration of olivine dominated rocks common to the Earth's upper mantle (beneath the Moho).

We have the remnants of such material not far from Goleta. In the Santa Inez Recreation area there are rocks of the Franciscan Formation. It is a tectonic mlange---a mixture of volcanic and sedimentary and metamorphic rocks that accreted onto what is now California around 140 Million Years ago!!!!

Wow!

Serpentinite often get squeezed up along fractures and faults that develop during plate collision.

Seefor some pictures:
figures

Good question!
Plate boundaries are sticky. If you have a collision between plate boundaries, bits of the two plates are going to wind up getting folded up, inserted into, or otherwise glued to each-other, and left in each- other. So, for example, if you have an oceanic plate colliding with a continental plate, pieces of the oceanic plate are going to get caught up in and entombed inside of the buckling that forms on the boundary, and some of those pieces are going to get stuck into the mountain belts that result from this buckling.

It gets even more dramatic when two continental plates collide with oceanic plate in-between them. Now you can get quite a bit of oceanic plate stuck in or on top of the joined continents.