Is it permissible for the agent to slaughter the constituent's sacrificial animal on the day of Eid if Eid in the constituent's country is one day later due to the difference in moon sighting between their two countries?

It is permissible for a Muslim to appoint someone else to slaughter his sacrificial animal (Udhiyah), though it is better for him to slaughter it himself unless there is a necessity. The time for the Udhiyah is after the Eid prayer; whoever slaughters before it, his sacrifice is not valid. If the timings differ between the country of the principal and the agent, then the decisive factor is the country of the agent. The agent may slaughter the Udhiyah after his Eid prayer, even if that is before the time for Eid begins in the principal's country.