Moving cheaply requires good timing, a handy amount of free labour (i.e. friends that own trucks and owe you favours or like you enough to help just out of the goodness of their own hearts), a little good luck, a keen eye on available real estate and good negotiation skills. Without all of these lining up, it's likely to cost a bit.
A couple of years ago, I got divorced, so I wiped the slate clean. Everything I couldn't fit into my beat-up 4-door sedan got thrown/given/donated away. I then drove 2500 miles to start building a new life in a different region of the country (US) where I had a job waiting for me. Maybe $500 in reimbursable expenses for gas and a night's lodging, then staying at a by-the-week motel until I scouted the area made the transition relatively inexpensive.
For many people, the cost of moving is directly proportionate to the quantity of stuff they own. I cut costs on moving by ridding myself of 95% of my stuff, and have been purchasing what I need in my new location.
Sure, not having a wife, 2.5 kids, and a dog makes these kinds of life changes much easier. But the point of having less stuff is still valid.
It's counter-intuitive, but often living frugally requires a higher up-front cost (such as buying quality, higher-priced goods that will cost less in replacement and maintenance in the long-run).
Also, never cut corners on a job, it'll save you time and effort in the longer term. Do it right, do it once.
Do your research up front, it may seem like a lot of time, but it'll pay dividends in the long run.
Have you taken a look at how expensive simply moving is? There's the rental truck or pod, there's first, last and security deposit for the new place. Sometimes there's a doubling up of rent. If you're moving to different cities, there's the price of gas or diesel for your moving truck. Not to mention the time and energy it takes to pack up and move.
Relocating isn't as easy as some people make it out to be....