I think it might make more sense to ask how transposition affects the pitches of a piece.
Usually a piece is transposed for a specific reason. The two most common reasons are to make it more comfortable for a singer or singers and to make it possible or easier to play with different instruments. In either of those cases, whether the piece is transposed up or down depends on exactly which instruments are going to be used and/or the ranges of the singers.
For example, it's not uncommon for a person with a high voice to sing a part an octave higher than the original, while a person with a low voice may sing an octave lower than the original.
When pieces are played on different instruments, it's usually called a transcription, because more than just a straight transposition is necessary. For example, the classical guitar version of "Leyenda" (often called "Asturias") by Isaac Albeniz involves transposition to a different key from the original as well as moving the lowest notes up or omitting them and moving the highest notes down. Also, the chords are arpeggiated. These changes are necessary because of the major differences between the guitar and the piano (the original instrument).
So in the end, a transposition to a different key might simultaneously involve some instruments playing higher and others playing lower than the original piece, or even some notes being played higher and other notes played lower on the same instrument, when compared to the original.