If you have less than 30% equity in the home you are moving out of and renting, the lender will require you to qualify with both the housing payments on the house you are renting and the new house you buy. I'm assuming your buying the new house to owner-occupy.
If you have 30% or more equity, if you can get a lease and deposit from a tenant on your house, you can count the rent towards your income.
Like any standard loan, you will need to have a lender check your credit, and you will need to send the lender your last 30 days paystubs, last 2 yrs W-2, and your bank statements to show reserves and the down on the new house. If you are self-employed, replace the paystubs and W-2's with your last 2 yrs tax returns, all schedules. With that the lender can fully pre-approve you so you are ready to make offers on a new home.
Financial records, credit scores, debt ratio will be reviewed qualified purchase with lender letter supported with sales offer.