It's not clear whetherIt appears that this change was not originally on the part of the Church first, orbut on the part of the governments who were generally responsible for the executions; and if the government changed before the Church there's a clear answer:on whom the Church stopped sanctioning these punishments because it knew they weren't goinghad to happenrely to carry out the secular sentences (the executions).
It As I recall, it wasn't until this century (I'm still trying to find a supporting document for this, going off my general recollections) that the Catholic Church stopped saying that the governments of historically Christian nations had a responsibility to hunt down and punish heretics in their nations; but governments had stopped supporting them long before.
The short answer, then, is this: the Church stopped sanctioning these punishments because it knew they weren't going to happen.
Note, now: the Church wasn't the party responsible for the first of these laws, and not every official or teacher in the Church agreed with them at first. Christianity itself, from the late first century to the early fourth, was not the most legal of organizations; and people were executed for belonging to it, not for falling away from it.
Laws imposing penalties on those convicted of heresy by Church tribunals were often created not as a result of organizationalby the Church's direct or indirect pressure on the part of the Churchgovernment, but by powerful rulers who had very strict personal interpretations of Christianity. The Roman emperor Theodosius I, with his two co-emperors, passed the first anti-heresy laws (apparently on his own and without any sort of political pressure). The 1911 Encyclopedia Britannica, discussing the Emperor Theodosius I in its article on heresy, states discusses these:
In particular, Bishop Priscillian of Avila had been engaging in practices and teaching things which a synod of local Hispanic bishops declared heretical. When Priscillian appealed to Emperor Maximus to regain his status (perhaps by putting pressure on the Pope?), Maximus had him and six companions executed on charges of sorcery. Priscillian's accusers were in turn excommunicated not only by Ambrose of Milan, but by the Pope as well. Ambrose and Pope Siricius did believe Priscillian to be a heretic, but also believed that capital punishment was somewhere between unnecessary and outright evil.