Yuri, I would suggest mentioning the work visa in an early email. I am sure there are and will be mixed opinions about this point of view. What I am sharing is my opinion and why I would choose this option. I prefer building trustful relationships, and am very sensitive about wasting other people's time, hence I think bring it up early is better.
Pros of bringing this up early:
A few employers are well aware of the effort of processing a work visa. They will be entering a discussion with you fully aware of your needs. You won't be wasting your time or a potential employer's time, going through the interviewing and engagement process only to find that your lack of employment visa is an obstacle to moving forward. (Some people will be upset to know that their time has been wasted because you weren't transparent or upfront with them about your lack of work visa and share their disappointment with you.)
Cons of bringing it up early:
Yes it will shut down potential opportunities. Processing a work visa does take additional work, cost, effort and justification and many people don't want to do extra work or incur that extra cost, especially for someone they don't know. Some managers might be willing to do that extra work, once they know you and the unique value you bring.
