Is it permissible to recite the post-prayer supplications (adhkar) at home, and if the answer is yes, is it preferable to remain seated on the prayer rug until finishing them, or is it permissible to move around?

The remembrances (adhkar) after prayer do not necessitate remaining in the prayer area. Rather, they can be recited while walking or riding. However, it is better for the Muslim to remain in their prayer area to recite them, due to the Prophet's (peace be upon him) saying: "The angels invoke blessings upon one of you as long as he remains in the place where he prayed, as long as he does not nullify his ablution, saying: O Allah, forgive him; O Allah, have mercy on him."

Ibn Nasrallah mentioned that reciting the adhkar while sitting is the apparent practice, but if one recites them after standing up or while leaving, they are also following the Sunnah (definition), and there is no restriction in that. If one remembers them after becoming occupied and then recites them, the special reward for them is apparent if it was soon after due to an excuse. However, intentionally leaving them and then making them up after a long time results in missing their special reward, though the general reward for remembrance remains.