Having just finished a contract in the investment banking arm of a Swedish bank, I can point to a range of frontline support and development roles which are going to be RELATIVELY outsource-proof. Telephony system support; desktop support; trading platform support; algorithm/decision support system development; basically anything that an investment banking trader is going to need in order to process a deal, although the specifics are going to vary by company.
Those traders are (a) paid a metric crap-ton of money, and (b) stressed by anything and everything from reflections on the screen to their coffee being too hot.
Having to pick up a phone and shout at someone in another country is not going to work - even someone in the SAME BUILDING, but on a different floor does not do it. They want someone in the same room (same room is not a small office, this is typically a trading pit with 250-300 people in it, so pretty big) who they can grab (sometimes literally), point at the problem and yell "fix it", before running off to chug down a couple of valium. If they do not get what they want, they bitch at their managers about how the company is not giving them the tools to do their job (make tons of money for the bank).
The typical overhead for one of those traders by the time their Bloomberg and Reuters real-time data licences plus software licences and broker fees is included, is well over $10,000 per month, and that is without accounting for their salaries, bonuses, IT hardware or software licences that are calculated at a corporate level - operating systems, Office package, databases etc. Overall, the "normal" monthly investment that the bank makes in that trader is over $100,000, so for the business, even when it is run by bean-counters or Harvard MBAs, whatever that person needs to increase their productivity, they get. So if the users say that they NEED an IT guy on the spot who can be at the desk within 20 seconds to look at and start working on the problem, that is what they get. If that IT guy is going to cost $80,000 - $100,000 per year instead of $15,000 per year for some guy called "Dave" with an Indian accent working out of Bangalore, tough shit, $65,000 is pin money.
Remote access from outside the bank is almost always flat out refused for security reasons, so while off-site support can work, third-party support has to be on-site and in person.
The only exception to the "local first" rule that I have seen, is the Bloomberg helpdesk. Those guys work remotely, and can be reached either by telephone (usually a call center somewhere in the same geographical region as the user) or by pressing F1 twice on your keyboard, which brings up a Messenger-style chat window. You type your question and it gets routed to a team that hopefully knows something about the specifics. Almost invariably though, when a trader gets a Bloomberg problem, it is batted off to the in-room IT guy who goes through the problem with the Bloomberg tech while the trader goes to pop another valium and get a massage to de-stress.
Basically, if you are someone who can turn Excel inside out, write high quality C++ code while being constantly interrupted, and solve whatever random crap problems come up while maintaining a calm demeanour and keeping the world's most stressed people from flying into hypertensive shock, then you have a job for life.
A typical day? Printer is out of paper; another printer is jammed, and has a full waste toner bottle; internet connection for user X is down; User Y has just sent a naked picture of client A's wife to client B by mistake; Murex overnight jobs have failed so the D3 trading platform has the wrong start prices and wrong date, so someone needs that escalating to the Murex support team to get it fixed; User Z has just got in and spilled coffee in his Bloomberg keyboard so the biometric login does not work; vendor A upgraded their app without telling anyone last night, so now all the users of that system cannot login; the row of desks 24-48 are completely dead, Christ knows why, although the building maintenance guys were checking the air-con for the computer cabinets last night; oh, and it is 07:45 and the various trading markets will be opening in between 15 and 45 minutes.
90% of the tasks you get in those roles are things that the average Walmart-level technician could handle in their sleep; 0.1% will require 5 years' experience and the ability to code Powershell scripts on the fly or edit and compile Linux modules; most of the rest are probably normal support calls, but with a much shorter reaction time needed. Oh, and every once in a while you get to put a baffled look on your face and ask one of the women working there "Just what were you trying to accomplish, sticking your underwear in there???"