We don't hear of most of the apsotles again, though there are traditions of where some of them went (e.g. Thomas to India). I don't know anything about Matthias.
A slightly side comment but Barnabus is referred to as an Apostle in Acts 14:14.
Acts 6 (1-6) describes how seven are appointed with special service, and the apostles laid hands on them to commission them. But they were not apostles and did not have all the authority of apostles. We can see this in that Philip had the authority to preach and baptise (Acts 8) but not the authority to call down the Holy Spirit. Peter and John, as apostles, had to be sent up from Jerusalem to do this.
We know that Paul hands on authority to Timothy and Titus whom he appoints to oversee churches, and he instructs them in the selection of people they in turn are appoint and pass on authority to.
So you, my child, be strong in the grace that is in Christ Jesus. And what you heard from me through many witnesses entrust to faithful people who will have the ability to teach others as well. (2Tim 2:1-2)
Paul refers to the laying on of hands on Timothy:
[SIZE=12pt]Do not neglect the gift that is in you, which was given to you through prophecy with the laying on of hands by the council of elders (1Tim 4:14)[/SIZE]
He instructs them in the selection of such people
For this reason I left you in Crete so that you might set right what remains to be done and appoint presbyters in every town, as I directed you, [SIZE=10pt]……… [/SIZE]For a bishop as God’s steward must be blameless, not arrogant, not irritable, not a drunkard, not aggressive, not greedy for sordid gain, but hospitable, a lover of goodness, temperate, just, holy, and self-controlled, holding fast to the true message as taught so that he will be able both to exhort with sound doctrine and to refute opponents. (Tit1:5,7,9)
The saying is sure: whoever aspires to the office of bishop desires a noble task. Now a bishop must be above reproach, married only once, temperate, sensible, respectable, hospitable, an apt teacher, not a drunkard, not violent but gentle, not quarrelsome, and not a lover of money. He must manage his own household well, keeping his children submissive and respectful in every way—for if someone does not know how to manage his own household, how can he take care of God’s church? He must not be a recent convert, or he may be puffed up with conceit and fall into the condemnation of the devil. Moreover, he must be well thought of by outsiders, so that he may not fall into disgrace and the snare of the devil. (1Tim 3:1-7)
Note how Paul instructs Timothy to guard the treasure that has been entrusted to him;
Hold to the standard of sound teaching that you have heard from me, in the faith and love that are in Christ Jesus. Guard the good treasure entrusted to you, with the help of the Holy Spirit living in us. (2 Tim 1:13-14)
He has previously left Timothy in charge at Ephesus with full authority over those there, particularly over those who had appointed themselves teachers, and false teachers at that.
I urge you, as I did when I was on my way to Macedonia, to remain in Ephesus so that you may instruct certain people not to teach any different doctrine, and not to occupy themselves with myths and endless genealogies that promote speculations rather than the divine training that is known by faith. But the aim of such instruction is love that comes from a pure heart, a good conscience, and sincere faith. Some people have deviated from these and turned to meaningless talk, desiring to be teachers of the law, without understanding either what they are saying or the things about which they make assertions. (1Tim 1:3-7)
Jesus gave the apostles the mission to take the gospel to the ends of the earth and he promised he would be with them to the end of the age. (Mt 28:19-20). That would not happen in their lifetime. Indeed it is not finished yet. That authority needed to be passed on to continue the mission.