You can request a distribution from another IRA and complete the rollover of the same cash or assets, generally in less time than it takes to complete a direct transfer. However, it is a reportable event, which means the resigning IRA custodian will report the distribution on an IRS Form 1099-R.
A distribution from your IRA is usually done either in the form of a check (if you've requested a liquidation) or as an in-kind distribution (if you requested re-registration of the assets). You then have up to 60 days to reinvest the funds or re-register the assets to your new IRA. If the funds or assets are not rolled over within 60 days, taxes and possible penalties would be incurred.
Important Note: A 2014 court ruling has changed the IRS position regarding how often IRA rollovers may be completed. Since 2015, only one IRA-to-IRA rollover can be completed. Please review the One-Rollover-Per-Year Rule for more information on this limit.
Learn more about rollovers at Fund My Account.