That may be so for some sailings but many, if they were sponsored or through an immigration scheme, had a passage to wherever the ship was leaving from included as part of the the package, in my case my gt grandmother and the toddler went to Plymouth from Co Londonderry. Then a fast trip via Teneriffe and the Cape to NZ, sometimes they stopped at an Australian port. These were the days in the 1880s where 'short' passage times were a point of difference and were avidly reported in the newspapers at the time. The ship my gt grandmother came out on, the Ruapehu, regualarly broke sailing time records.
I have had ancestors leave from the south or west of ireland, at this time, but they were mainly going to the US.