Although your question is not clear, I’d try to answer it from whatever I made of it. According to me, you are trying to ask whether the chat conversations are seeing their end today, and I would like to disagree with you.
As per my personal experience, chat conversation is an easier way of communicating and allows you to keep a conversation ongoing even when you are busy. Chat conversations, started with Blackberry messenger and Hike, and reached the heights of popularity with Whatsapp messenger, Wechat, and Facebook messenger. Even Instagram gives you the feature of chat conversation.
We all know that we are not always confident enough to converse with some people face to face, or even on phone. Emails, on the other hand, are way too formal and the replies can be delayed by days on emails. Keeping in mind all this, chat conversations is an easy way out to communicate without having to come out of your comfort zone or without waiting for long.
With all these advantages, I don’t think the trend of Chat conversations is ending anytime soon.
Now the second think that I made of your question is that you want to ask how to end a chat conversation, without sounding rude. Well, this is another advantage of chat conversations –you don’t owe any explanation to end the conversation, and more than often you don’t even have to come up with an excuse. All you have to do is say “TTYL” (talk to you later) or “gotta go”, or even “mom is calling”. Trust me, the person on the other end also understand that there is no obligation of continuing the conversation over text messages.