I've received moving expenses three times in my career. The first was when I moved to Chicago for my first job. I think they gave me $2000, and I think I had to ask for it. I had no idea how or what I would be using it for. It was just kind of free money.
When I moved to Europe, it was a little more complicated, naturally. They paid the airfare for me and my whole family, another $7500 to move my stuff from Chicago, and fronted the money for the downpayment on my apartment (local laws required 3 months, so it came to about $9000).
When I came back to the States, my current agency put me up in corporate housing for a month, but typically they would have only done it for a week (I was their first transocean move). I paid $5000 to move all my stuff from Europe to America, but negotiated to have the agency move my furniture from our old house in Chicago to our new place. (We'd rented our house in Chicago and kept it furnished.)
I think the key is to figure out what you need and then tell them and negotiate from there. Go on pods.com and get a quote on moving to your new city. If you're driving, figure gas money, and maybe even one night in a hotel. Just make your plans as if you were paying for it yourself, and ask the agency if they know any way to make it less expensive. I gave my current agency an estimate from pods.com, but they had a corporate account with a moving company that was less expensive, and I was fine using them.
When I moved to Europe, the agency offered $5000 to move, which I didn't think was going to be enough. I wrote to them and said, "Since you've flown me business class twice for interviews, I assume you'll fly my entire family business class when we move. How about you fly us all coach, and give me half the difference in moving expenses, which will be about $5000." The CEO wrote back and said he had planned to fly us coach, but could offer an additional $2500 to move, but no more, which was totally fine with me.
Be honest. Be fair. Negotiate for as much as you can. And realize that however much they're giving you is probably coming out of someone's Chirstmas bonus or the office holiday party.