Hi folks.
I use DeltaChat on Android, iOS and Linux. There is no synchronization of sent mails between the devices.
Also, there is a problem with the background synchronization at the iPhone. The app simply didn’t run at the background. At the preferences the option for background service is activated.
Hi Marko 
welcome to the community.
If you use multiple devices, it makes sense to enable “send copy to yourself” everywhere, so you see your own messages on all devices: FAQ - Delta Chat
iOS often stops Delta Chat from running in the background, and only wakes it up occassionally. Note that Delta Chat doesn’t use Google Cloud Messaging (GCM) or the Apple Push Notification Service (APNS), because this leads to user tracking and central control which Delta Chat aims to avoid as much as feasible.
However, that Android and iOS kill apps running in the background is a problem for many legitimate apps. For more information, see dontkillmyapp.com.
It works now, thank you. The option “send copy to yourself” was not activated. The fun fact is, that I use a degoogled android phone (CarbonROM) and there DeltaChat works perfectly at the background.
My daughter uses a Huawei P10 with PlayServices and there DC is constantly killed at the background. I found no working solution to avoid this.
Yeah, I also use a custom ROM without google and it works perfectly.
It’s really about freedom - Google and Apple want developers to use their centralized push notification systems to gather metadata, and claim it’s about “saving resources”. The energy and network resources of keeping one listening connection open are of course negligible. But they want to pressure developers into this. https://dontkillmyapp.com/ is really worth a quick read, if you’re interested in the topic.