Hybrid cryptosystems, as @kelala has mentioned, are used in practice in favour of public-key encryption systems. It's not a particularity of Elliptic Curve vs Integer-Group based cryptosystems: it's rather that bit-twiddling (symmetric-key) cyphers are much faster than algebraic (public-key) ones.
So, what we do is to create a symmetric key and use a public-key scheme to encrypt it in the KEM/DEM paradigm (and also 2) and then use a symmetric scheme to actually encrypt.
Also, keep in mind that even if you do encryption wholly in an Elliptic Curve, you probably need a hash function at some point that maps the text message (a sequence of bytes) to a field element or a curve point itself.
You can look up the work of Icart, Fouque, Tibouchi, Coron, Maurer and others, for examples of how to do this (depending on the curve family you're targetting)