Applying for long term visit during PhD
Ok, lets look at the core issue here (in your and my mind at least): publications and future paths; and then your question.

Lectureship in the Horizon?: This is a touchy subject, and I'm sure academics get divided on this one. However this is a fact: how many postdoc students did you witness with a weak publications who could get a good lectureship position. So the conclusion here is you need good publications to survive and progress as an academic.

Postdoc Isn't the Holly Grail!: This is another one of those time wasting adventures of people who did not think right about their abilities. What environment are your looking for? If you are looking for a perfect environment so you can keep publishing new articles every couple of months; let me tell you it does not exist. Rather, your ability on writing your ideas and contributions are only in your hands. Now if for some very good reason you couldn't publish much during Ph.D. then your new postdoc position could be another story; however if you did your Ph.D the best you could and you still struggled to publish 3 good papers; then let me tell you do not waste your time and energy on doing a postdoc.

Conclusion on the first two points: If you know why you want to continue research after doing a Ph.D., you better understand the importance of publications. If not, you are just wasting your time. 

Visiting a Group: Of course you can visit a professor and his/her research group; however two things here: 

1.Who Do You Know?: If you know the professor then send him/her an email and request a visit. If not who your supervisor know, so you can go and visit? Other than that, unfortunately people simply ignore those emails.

2.What Will You Provide?: What will you provide to the group you want to visit. Are you going to a group lets say in Glasgow, just to see Glasgow or you want to go there to collaborate? What you want to provide to them? Will you create an unfriendly environment for them and moan about your Ph.D. thesis or you go there for research? You see what I'm trying to say here? This again goes back to the first point that people usually can visit another group, when the professor knows the person first hand or second hand but from the very good source.