I never actually applied. ... I had my resume posted on Career Builder and was contacted by a recruiter at a staffing agency. From the time the recruiter contacted me to the final round of interviews, it took about 2 months. I had a brief conversation with the recruiter about what I was looking for in a career and a company, and she thought it would be a good fit, so she sent me on to an initial interview with the Director of Operations. That day, I received an invitation to attend the final round of interviews with a group of other candidates.
The final round of interviews entailed interviews with several VP's and Directors. The day also included a visit of their warehouse and a visit to a nearby store with two district managers who have both been in position for several years. Ask tons of intelligent questions and make a great impression!
ALDI paid for hotel/travel for the final round of interviews, but not the initial meet & greet type interview. Overall it was a very smooth and informational process.
If you are open to relocation, prepare for the interview, and fit the personality of the interviewer, you shouldn't have any problem at least getting to the final round of interviews. Like I said before, ask a lot of questions - most of my interviews were me asking them questions rather than them asking me!
Just be yourself; explain your background and your interests with the position & company - It is a conversational type interview. They did ask a few questions, mostly about accountability and the type of leader you are.