The basement rocks of the continental crust tend to be much older than the oceanic crust. The oceanic crust can be from 0-250 million years in age, and is usually thinner (10 miles or so) and composed of basaltic rocks. Continental crust is older because continental crust is light and thick enough so it is not subducted, while oceanic crust is periodically subducted and replaced at subduction and oceanic rifting areas.
The basement rocks are often highly metamorphosed and complex. They may consist of many different types of rock - volcanic, intrusive igneous and metamorphic. They may also contain fragments of oceanic crust that became wedged between plates when a terrane was accreted to the edge of the continent. Any of this material may be folded, refolded and metamorphosed. New igneous rock may freshly intrude into the crust from underneath, or may form underplating, where the new igneous rock forms a layer on the underside of the crust. It is said that the majority of continental crust on the planet is around 1-3 billion years old, and it is theorised that there was at least one period of rapid expansion and accretion to the continents during the Precambrian.
Much of the basement rock may have originally been oceanic crust, but it was highly metamorphosed and converted into continental crust via a series of events. A typical pattern is as follows. It is possible for oceanic crust to be subducted down into the Earth's mantle, at subduction fronts, where oceanic crust is being pushed down into the mantle by an overridding plate of oceanic or continental crust.
When a plate of oceanic crust is subducted beneath an overriding plate of oceanic crust, as the underthrusting crust melts, it can cause upwelling of magma that can cause volcanism along the subduction front on the overridding plate. This produces an oceanic chain of volcanoes, like Japan. This volcanism causes metamorphosis of rocks, intrusions of magma that produce rocks such as granite, and thickens the crust by depositing additional layers of rock from volcanoes. This tends to make the crust lighter and thicker, as a result making it immune to subduction.
Oceanic crust can be subducted, while continental crust cannot. Eventually, the subduction of the underthrusting oceanic crust can bring the volcano chain close to a continent, and collide with it. When the overriding plate collides with the continent, instead of being subducted, it is accreted to the edge of the continent and becomes a part of that continent. Thin strips or fragments of the underthrusting plate may also remain attached to the edge of the continent causing those fragments of oceanic crust to be wedged and tilted between the converging plates. In this manner, continents can grow over time as new terranes are accreted to their edges, and so continents can be composed of a complex quilt of terranes of varying ages.
As such, the basement rock can become younger going closer to the edge of the continent. There are exceptions of however, such as exotic terranes. Exotic terranes are pieces or fragments of other continents that have broken off their original parent continent and have become accreted to a different continent.
Also, many continents can consist of several continental cratons - blocks of crust built around an initial original core of continents, that gradually grew and expanded as additional newly created terranes were added to their edges. For instance, Pangea consisted of most of the Earth's continents being accreted into one giant supercontinent. Most continents indeed have several continental cratons, in other words they are made up of the accretion of many smaller continents, such as Asia, Africa, and Europe.