There really are not constitutional lawyers like there are tax, bankruptcy, patent, et cetera lawyers. Rather there are some lawyers who try a lot of cases with con law issues. There might be a few public sector lawyers who only deal with con law issues but that would be the extent of it.
Further when you ask what a con law lawyer does you are asking to broad of a question. Thats the equivalent of asking what a criminal defense lawyer does. A criminal defense attorney may do white collar crime, petty crimes and DUI, rape, kidnapping and murders. They may have done all of those at some time in their career but its doubtful they currently do all of them as one takes specialization (white collar) and a criminal law laywer doing murder defenses and making good dollar will be unlikely to want to take petty crime cases.
Same with constitutional lawyer, the could do civil rights (substantive due process and equal protection), criminal (4th, 5th, and 6th amendment), property (5th amendment eminent domain) or they may be very unusual and be an expert like something like separation of powers or federalism (though usually those types would be law profs).
If you get anything from this post, take this: There really isn't such thing as a constitutional law lawyer (with a few rather narrow exceptions)
Also this was posted on the wrong board, the meta discussion board is to discuss the law school discussion board, whats wrong with it, how it can be improved and so on, this should have been posted on the general topic board.