The descendants of emigrants often long to heal the generations-long breach in their family by researching forward to find living relatives. But it is one of the most difficult tasks possible, going right against the grain of time. So:
Chronicles of extended periods are the basics of such research, and the single best source is the Valuation Office collection of Revision books, which detail all changes affecting those liable for local property tax. In the Republic, they cover the entire period of about 120 years between the original Griffith's and the abolition of the rates in 1977. The 26-County books are all still only available at the Office itself (see valoff.ie for details). For Northern Ireland, the books are online at proni.gov.uk, but only come up to the 1930s, when a full re-valuation took place.
The other main chronicles are annual urban street directories, useful mainly for Dublin and Belfast, the latter online at Proni, the former best accessed at the Gilbert Library in Pearse St. Electoral lists, in theory revised every year by the relevant local authority, can also stand as proxy directories. The best collection is for Dublin, at dublinheritage.ie and again in the Gilbert.
Reconstructing entire families is the first step in following indirect lines of descent, and being able to search birth records by mother's maiden name is an essential tool. The state birth indexes from 1903, (soon to be online at irishgenealogy.ie), the Northern online registers at https://geni.nics.gov.uk and the transcript databases at rootsireland.ie are the main sources.
Records associated with deaths are also useful, particularly secular burial records that give next of kin or multiple interments in a single grave. Examples are Glasnevin in Dublin (glasnevintrust.ie) and local authority records (kerrylaburials.ie).
And don't forget newspaper death announcements, a staple of every Irish funeral since the 1940s, which often have long lists of grieving relatives. The richest sources (it grieves me to say) are The Irish Independent and The Irish Press, both online at irishnewsarchive.com.