The problem of having multiples Browser each of them have different protocols and different ways each Browser manipulate the same Html.SharePoint Designer have a option to preview your SharePoint Page, but by default is associate to Internet Explorer but you have the ability to associate the multiple Browser to preview the page in a very way.
To add new Browser in your "Preview in Browser" option you will need to select "Ribbon > Preview in Browser > Edit Browser List".
To add a new Browser reference you need to select option "add" and "Name" and "Command" associate to your Browser.
Here some references i use do add some Browsers:
"C:\Program Files (x86)\Google\Chrome\Application\chrome.exe
"C:\Program Files (x86)\Mozilla Firefox\firefox.exe"
After made the reference to each Browser .exe you can use the Option on SharePoint Designer "Ribbon > Preview in Browser" and select the Browser you want to display the SharePoint Page you change and evaluate the behavior.
Below you have a link from Microsoft about the Browse support SharePoint Sites:Plan browser support (SharePoint Server 2010) http://technet.microsoft.com/en-us/library/cc263526.asp Hope you like it.