In the Wikipedia list of legislative bodies which use staggered elections only upper chambers of national parliaments can be found.
Why? Is there a reason why staggering the elections for the lower chamber would be a bad idea?
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There is a tradition that the lower chamber should reflect the current democratic will of the people, and the Higher chamber should be a more reflective body, able to "wisely" consider and advise legislation. To this end it makes sense to hold staggered elections in the upper house. It means that the composition of the upper house cannot change rapidly and members of that house will have long terms of service and so do not need to be focussed on re-election. They can consider if a bill is "good" without having to think if it is "popular".
The lower house, on the other hand, is a reflection of the democratic will of the people. This is why it is the lower house that usually has the final say on finance bills (no taxation without representation).
When countries design bi-cameral chambers, they often look for existing models of successful democratic countries. It is clear that many of the countries that use staggered elections do so following the model of the USA and the staggered elections for the Senate.