Is there a one-year gap between Muawiya's death and the event of Karbala, or did Imam Husain leave Madina within 6-7 days of Yazeed's rule since both events took place in 680 AD?
Based on historical sources, there is about a 6-month gap between when Yazid took power, and when Imam Husayn (A) left Mecca. There was also some time gap between when Mu'awiyah died and the news reached Medina (I don't know how long it took for messages to travel then).
Here is a timeline based on what is related:
Rajab 680 - Mu'awiyah dies
28 Rajab 680 - Imam Husayn (A) leaves Medina
3 Sha'ban 680 - Imam Husayn (A) enters Mecca
Dhu al-Hijjah - Imam Husayn (A) leaves Mecca for Iraq