First keep in mind that each visit in a different browser does not have the cookies from the previous visit in the previous browser. This means that target treats the visitor as new in each browser and has to re-evaluate what campaing and experience they qualify for.
If you have two campaigns (Campaign A and Campaign B) competing in two mboxes (mbox1 and mbox2) it is possible that the same end user will see Campaign A in mbox1 in Firefox and Campaign A in mbox2 in Google Chrome.
Understand that this is working as designed. If you want to prevent this behavior you will have to restrict the campaign to only fire in one mbox.
It is possible that the original browser you tested in has old cached information and is displaying the experience and campaigns the way they were configured prior to the last changes. In this scenario If you have a campaign (Campaign A) configured to fire in mbox 1 and you first tested in Firefox and then re-configured the campaign with a new location of mbox 2 and then tested in Google Chrome then Campaign A will show in mbox 1 in Firefox and mbox 2 in Google Chrome. Until you clear the cache.
Clear the cache in the original browser and you should see the campaign appear in the same location.
It is possible that someone has changed the location the campaign is targeting mid test. This will cause your experience to differ if you are testing on multiple browsers depending on what information each browser has cached. (pre or post change)