The prefix per-, when used with verbs of movement, means a direction through or across something.

pereiti - to go/walk through or across smth
perbėgti - to run through or across smth
pervažiuoti - to go (by a transport) through or across smth
perskristi - to fly through or across smth
perjoti - to ride through or across smth
perplaukti - to swim through or across smth

Aš pereinu tiltą. I walk across the bridge.
Aš pereinu per tiltą. I walk across the bridge.
Tėvas perbėga gatvę. Father runs across the street.
Tėvas perbėga per gatvę. Father runs across the street.
Laivas perplaukia kanalą. The ship swims across the canal.
Laivas perplaukia per kanalą. The ship swims across the canal.
Asilas perjoja upę. The donkey rides across the river.
Asilas perjoja per upę. The donkey rides across the river.
Paukštis perskrenda jūrą. The bird flies across the sea.
Paukštis perskrenda per jūrą. The bird flies across the sea.
Autobusas pervažiuoja duobę. The bus drives across the pit.
Autobusas pervažiuoja per duobę. The bus drives across the pit.

Since both have the same meaning, the preposition per serves as an emphasis when used together with prefix per-.