No, not all the zoathids will display the same sign. no.
To tell you the truth, not all the zoanthus spp. or Palythoa spp. (the most common zoanthids in the trade) will display the same signs.
Sometimes even the same species of zoas, as many here have posted before, could be affected differently when moved from different tanks. Sometimes even from same mother colonies.
They will show signs of bleaching when stronger than need light is offered. That could happen when temperature is higher than the acceptable average also. Or both!
Basically you need to keep your eyes open and aways start from the bottom of the tank with new zoas, when possible. The main reason for that is because 100% of the zoas you'll get will need to adapt to the new tank in every aspect, so that's the safest way to go at least in regards to the light.
Zoas are not that difficult to adapt to new light sources in most cases, and if you start from the bottom of the tank it should be fine as a basic suggestion. The time of adaptation is normally a month, I would say, depending on how different the light sources are (old and new lights).
Keep in mind that if your light is much more powerful than the light they were under before you'll need to shade them and gradually bring them out for your type of light, as we all should know by now.
Careful when you change bulbs too. They can be harmed with sudden changes.
In regards to getting brownish, that normally means low light and/or too much nutrients, as we already have observed too.
All that needs to be cared for.
There is no rules nor a magic portion you can lay on.
Every time you have a new zoa there will be a new beginning of observation period that will vary for each individual polyp or colony.
Hope that helps a bit.
Grandis.