This is the 14677th most frequent Russian word.
пересекая
"Пересекая" means "crossing" or "intersecting."
Here, the word 'пересекая' is the adverbial form of the verb 'пересекать', relating to the process of crossing something while in motion.
Он шел через парк, пересекая тропинки.
He was walking through the park, crossing paths.
In this sentence, 'пересекая' acts as a participle describing the simultaneous action of crossing the border while presenting passports.
Пересекая границу, путешественники предъявили свои паспорта.
Crossing the border, the travelers presented their passports.
The usage demonstrates the action of crossing as being a significant and planned part of the activity in context.