The reason we kiss the Cross in procession is very simple and very historical.
When the Roman empress St. Helena, Constantine’s mom, dug up Jesus’ cross and some others that were lying around Golgotha, she figured out which one it was by testing which one had healing power. (I seem to recall that she had some local cripples brought out, or some such, and had them touch the crosses she’d dug up in turn. The one that made them get up and walk was obviously the right one.)
Naturally, after seeing this, everybody and their uncle Bobbius wanted to touch or kiss the Cross. So that’s what they did – they lined up and touched or kissed it. And that’s what they continued to do in Jerusalem for years and years. Other places made their own pretend crosses, and did the same thing, as a sort of pretend pilgrimage to Jerusalem. (Much like the Stations of the Cross, which is a sort of mental pilgrimage to the Via Dolorosa.)
Eventually, they stopped having processions with the actual cross of Jesus, though, because somebody cut off a big chunk of the Cross while pretending to touch it and then carried it off in his mouth while pretending to kiss it. Usually, the only people who got pieces of the True Cross were foreign dignitaries whom Constantine or the later emperors were trying to make pacts with, so this was not just sacrilege; it was also a national security issue. It was like stealing the crown jewels, if the crown jewels were also incredibly holy. So that was the end of that.)
(And that’s why there are so many splinters of the True Cross kicking around – though they amount to less than a quarter of the volume of the original cross, according to the obsessive Victorian guy who studied every single splinter still in existence. It was a sort of Byzantine diplomatic gift, and the empire just dealt with an awful lot of people over the course of eleven hundred years.)