Depending on the location, there is a scheduled day for weekly cleaning. Flowers are removed which possibly includes the "cones" they are in. There are also geese in the area who drink water from those "cones". Those geese may be the ones moving the flowers to get to the water.
Talk to the office for this issue .... I think the one you need to talk to .... her name is Monica
Are you talking about out in the grass, the holes for flowers in the headstones? I think every Thursday (I could be wrong on the day) Holy Cross mows the lawn so flowers might be moved? Just a thought.
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