In my opinion: always at the end.
Some people might not want to give their email address and might just stop there. Also, they do not know your survey yet, thus might not be sure if they want to take part the second round.
So I would ask for the email at the end.
Just an example from market research practice: We often do interviews in the home of people, to not only get responses to questions, but to also observe and see how people live. That can give you a lot of insights. We usually also want to take pictures to include into the report. We are not starting with taking pictures. We will sit down together talk about the topic, create a nice atmosphere, show them that we are not some weird freaks, but honestly interested in their opinions, needs, desires, etc.
Only then we'll ask them to have a tour of their home, see the kitchen, see the bath room (depending on the product category we are researching). Of course they are informed before, that we want to film/take pictures, during recruitment. But it would still be weird getting to their home and straight go for the photos.
After the team has been with them for 1, 2, 3 or sometimes 4 hours, they are generally much more relaxed and you are basically "a friend" already.
Same for your survey: They don't know what to expect, who you are, what you want. So go easy on them at the beginning. If they enjoyed taking part in the survey and the topic is relevant to them, they will give you their email address. If they did not enjoy it or it was not relevant to them, they will not answer anwyay.
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Note: I answer at this forum in my spare time, I'm not a LimeSurvey GmbH employee.
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