Well, I suppose the term "mail-order bride" is a bit of a misnomer, because you don't just chose a girl from a catalog and then she flies over to your home country, rather such sites are just international dating sites. However don't let the fact that it is online dating deter you, the response rate with the girls overseas are completely different to what you would expect from girls living in Westernized countries.
On a slight tangent, feminism, female entitlement and hypergamy are rampant. In North America dating sites favor women, you get a lot of unattractive, over-weight woman who are punching way above their weight, because most men are sex starved nowadays and are willing to date someone on the short-term, lower than them in the looks department just to get some regular sex. There are a whole heap of other reasons, but suffice to say that unless you are in the top 20% in the looks department as a male, you chances of having any success is negligible.
But back on to the topic, you converse with these women just like you would on any other dating site. You send them letters and expressions of interests, and wait and see who reciprocates. When you connect with a few of them, you then make plans to travel to their country of origin to see if there is any chemistry between you two.
Now of course this endeavour comes with risks. The main one is being scammed by the dating website or the girl. Never use sites that are pay-per-letter, meaning sites that charge you money to read each letter the girls sends you. These are are scam sites where the whole objective is to get you to spend a whole heap of money conversing with girls before they suddenly disappear. Regardless of what the excuse the website comes up with (that they act as the middleman to prevent you from being scammed is the common one) the sole objective is to take your money and run. You also run into women in their home country who are only interested in getting you to spend money on them, going to fancy restaurants or getting you to buy them expensive clothes. Be careful, such girls are known as “pro-daters” and if you meet one, cease contact with them and move on. The other ones you need to be careful of are the most well known, the visa vultures. My only advice on this is to trust your instinct and judgement, your intuition will let you know if the woman are genuinely interested in you, or just want citizenship.
It will probably be the best thing you have done. The women tend to be more feminine, less aggressive and more down to earth in what they expect in a partner. In the West if you're not rich, ridiculously good-looking, tall and meet the conventional Hollywood notions of beauty, your chances of finding a partner is dismal.
Source: I have been married to a girl from the Former Soviet Union - Azerbaijan - for 4 years now, we currently have 1 daughter together and am expecting another soon.