Is there a Quranic verse related to the dropping of punishment from a perpetrator of a crime due to the expiration of the period for filing a lawsuit against him?

There is no verse that stipulates the forfeiture of rights by prescription. Rather, Islamic rulings affirm that established rights are not forfeited.

Thus, if someone commits a crime related to the property or honor of a Muslim, the claim for it does not lapse with the passage of time. Instead, forgiveness must be sought from the right-holder or the property returned to them, as the Prophet, peace be upon him, said: "Whoever has wronged another concerning his honor or anything else, let him seek his forgiveness today, before there are no dinars or dirhams."

If the crime concerns a legal punishment (حد شرعي) like adultery or murder, the punishment must be applied even if the perpetrator seeks refuge in the Sacred Precinct (Haram (definition)), as mentioned by Sheikh Al-Kharashi Al-Maliki.

As for highway robbers (muharibun), if they repent before they are apprehended, their right to Allah Almighty is forfeited through repentance, as Allah says: "Except for those who repent before you apprehend them. And know that Allah is Forgiving and Merciful." (Al-Ma'idah: 34). This removes the obligation of execution, crucifixion, amputation, and exile from them.

However, the rights of human beings are not forfeited from the repenting highway robber; rather, they are still liable for retribution (qisas) in cases of life and injuries, and for financial compensation (gharamah) and blood money (diyah).