The most common way to get into an Alamo Ring is Allemande Left in the Alamo Style. This starts like a regular Allemande Left, but instead of dropping left hands and facing their original partner, they keep turning and take right hands with their partner.
Because an Alamo Ring is like a wave, it is possible to do Swing Thru from an Alamo Ring. Each pair of dancers holding right hands turns 1/2 by that hand, and then each pair of dancers holding left hands turns 1/2 by that hand. Since an Alamo Ring has no "ends", every dancer can do both parts.
Similarly, it is possible to do a Run. Because every dancer has another dancer on both sides, the caller will typically say which direction to run, e.g., "Boys Run Right". (But if no direction is specified, the default is right.)