The economic sectors that can be easily outsourced are the sectors where we see the most stagnant wages. Call centers, manufacturing, low level IT jobs, human resources and so on. Jobs that are not easily outsourced have seen the bulk of wage growth. Example: healthcare, higher level IT jobs, financial advisers and so on. Automation and container ships have more to do with the decline of manufacturing jobs than anything else. Illegal immigration does have some impact at the bottom of the wage scale, but the biggest factor in terms of flat income growth is flat productivity growth. No one has came up with anything that significantly increases worker productivity in nearly 20 years now.
For example, in the 1990s illegal immigration was very high, yet the median household income went up every single one of the Clinton years, poverty rates went down every single one of the Clinton years (reaching some of the lowest levels on record), and there was significant wage growth at all income demographics. That is the only time that has happened in 40 years. Now, while some people would credit the Clinton Administration with that, they are not the primary reason for it, but rather it was the IT revolution of the 90s. For example, in the late 80s, had you walked into an auto parts store, the guy behind the desk would have had to look up a part in huge book and then call around warehouses to see if he could find it assuming it wasn't in the back (with each of those warehouses looking through inventory books). By the early to mid 90s, every parts chain in the country was going to digital inventories where they could find a part within a couple of keystrokes. The same is true with company financials, business services, business operations and so on. The 90s saw the biggest jump in productivity since the 60s and as a result wages grew and the economy was stronger than at any time since the 1960s (before or since the 1960s for that matter). This was all despite rampant illegal immigration in the 90s.
For us to see strong wage growth again, we are going to have to figure out a way to get strong productivity growth again.