I heard Asleep at the Wheel's first album in 1973, and it began for me a deep dive into Western Swing, beginning with Bob Wills and the Texas Playboys, who were undoubtedly the premier practitioners (Spade Cooley, out in LA, had an excellent band as well, but the Playboys really defined the genre). There were a slew of very-good-but-less-successful bands as well during the heyday of Western Swing.
Some might look on Wills and his boys as simply white players playing the swing music of the time, but it was much more than that. Oklahoma and Texas were a remarkable potpourri of cultures and styles - you had not only the blues, but Mexican, German, and Hillbilly music. And even though we rightfully credit Charlie Christian with groundbreaking electric guitar, the Playboys were doing similar things at the same time, but with electric guitar AND steel guitar AND electric mandolin... not to mention incredible fiddling. Fortunately, Bob and the boys left us a great deal of wonderful recordings.
In regards to the bass stylings of the Playboys: I would call it half-and-half 2/4 and 4/4 (walking), not including waltzes. For instance, "Take Me Back to Tulsa" and "San Antonio Rose" were 2/4, while "Roly Poly" features a fast walking line. Sometimes the bass would break into a 4/4 on an instrumental break.
The piano stylings, with a lot of stride piano, dictate the bass parts. Have a listen (and look) to this GREAT film clip of
"Ida Red" from the late 1940s. On the vocal choruses, the bass part is 2/4, descending and ascending with the piano's left hand; on instrumentals, the bassist walks four to the bar. It's really a jazz styling of a song we might THINK of as having two chords, but in fact there are six or seven chords being played, with a chord change on each measure, ala guitarist Freddy Green of the Basie band.
You will find the walking lines more typically on the medium-tempo tunes like "Bubbles in My Beer" or "My Confession." These tunes are the precursors to the shuffles of the mid-fifties honky-tonkers like Ray Price.
There's nothing really unique about the Playboys' walking bass lines; any decent bassist of the time could do that. Due to the limitations of recording at the time, it's also hard to discern the Playboy bass lines. A better place to learn that style might be from the early recordings of Count Basie's band, or Jay McShann, or the aforementioned early Ray Price. For that matter, just about any of the great swinging R&B from the late 40s or early 50s will feature walking bass, and it's not significantly different from Western Swing. An album that I wore out listening to, and learned a lot of walking bass from, is
"Big Joe Turner - The Boss of the Blues sings Kansas City Jazz."
To walk well, you will need to learn some theory and the concept of chord substitution, e.g., substituting ii-V7 for IV in a blues. I learned to walk well enough that once I
didn't get a job with a blues act, because they thought me "too smooth."
During the 1970s, I had the pleasure of warming up for Asleep at the Wheel three different times (once, they tried to hire our sax player). Ray Benson has to be one of the hardest-working men in music, and his group has always been the real deal.