This would depend on the pilgrims religion. For instance.....
A Muslim ( being a person of the Islamic faith), will make a pilgrimage (called Hajj) to Mecca (Makkah) at least once in their lifetime, as long as they are physically and financially able too. The annual Hajj begins in the month of Dhul Hijjan, which is the 12th month in the Islamic (Luna) calender year. A pilgrimage to Mecca is one of the "5 pillars of Islam".
The pilgrimage went on for over a year
Islam requires a pilgrimage for every able man to Mecca. This pilgrimage has to happen at least once.
Yes. Cyclones can happen at any time of year.
Tornadoes can happen at any time of year but they are most common in sprind and early summer.
December
A lunar eclipse can happen at the time of any Full Moon, but no more than about 5 times in a year, max.
Usually winter time
The believers were going to make a pilgrimage to their holy land.
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They can happen at any time of year and any time of day, but the greatest number happen in the moths of April, May, and June.
It could happen any day of the year.