Share your experience!
I might have the same problem. And I have trasferred 8000 sms from my old Z3 Compact with the app Mybackup pro.
Messenger app opens fast. But when I sending sms the app is processing the sms very very slowly.
I have tried this solution also. "Try go to Settings -> Apps -> tap the 3 dots -> tap on Reset app preferences"
..... and my phone is Z5.
Could you please let me know if the same thing happens in other messaging apps like Google Hangouts for example? You can switch to hangouts from Hangouts -> Settings -> SMS -> SMS disabled.
I have tried Google Messenger and Hangouts now.
Hangouts are slow just like Sony's Messenger.
Google Messenger are processing and sending sms fastest at the moment. Not as fast as the Sony's Messenger on my old Z1 and Z3 Compact. I'm going to use that app for now and test Sony's Messenger after the next firmware and app update.
I have the same problem. also, one message just stood spinning for three days, I dont think it was sent, then i deleted it. and it is an on and off thing as well, sometimes its bad, somtetimes its not.
so i cant trust the messages to be sent, and not that i get the ones that are sentto me. thats bad.
this makes me quite a bit diappointed to be honest, I just dished out much moneye on something i expected to be the best phone on the market, and then theese problems arise 
Graffin, it seems to be ok when I deleted all my old messages/conversations. it took some time doing so though. i just find it odd that my old iphone 4s could handle all those messages just fine, whil my new sony xperia z5 compact couldnt. seems as if the software could be tuned better. in fact, it seems many have this problens, so imho Sony should discourage people form transferring their old messages from their old iphone units. it just created a mess.
Same problem here :S
The messaging app and hangout are really slow and takes a long time to send, receive and load the messages.
This is really dissapointing, my old Z1 didn't have this sort of problem before.
I mean seriously sms is like one of the most basic / oldest function a phone can do, how did Sony seriously mess it up ¬_¬.
heres to hoping for an update asap
I've received some more information about this.
If you have a very large number of stored messages in the Messaging application you may experience that the application becomes slow. Work is ongoing to solve this, and there will be improvements in future software updates.