Physical endurance can be an issue for woodwind players, most often manifesting as fatigue in the muscles of the embouchure. But I think in most cases tired facial muscles are a symptom of a more fundamental problem.
The muscles used for forming woodwind embouchures are small and finely-tuned for precise movements, such as in speech and in facial expressions. This also makes them well-suited to the fine control needed for woodwind playing. But those muscles are not really adapted to feats of strength or endurance.
Tired and sore embouchure muscles lead to additional problems, such as compensation by clamping down with the larger, stronger jaw muscles, which sacrifices control and causes woodwind players (especially reed players) to bite into their own lips. (As a less-experienced player, I thought of those raw, swollen, and eventually calloused spots in my lips as signs of dedication to practicing. I don’t have those spots any more. In many cases, the need for some kind of cushion or dental appliance over the teeth when playing is a sign of unnecessary biting.)
Woodwind players should be doing most of their physical “work” with muscles that have strength and stamina. The “core” muscles of the torso have both: they are an integral part of posture, balance, and virtually all gross motor activities (walking, jumping, lifting, sitting, standing, and many more). The core muscles are also the muscles of breath support, which is arguably the most crucial, foundational aspect of woodwind playing.
Powerful breath support takes a huge burden off the facial muscles. For example, it stabilizes pitch, reducing the need to “lip” notes up or down (which is a less-effective technique anyway); it strengthens and solidifies tone, reducing the tendency to “control” the tone (poorly) by biting or squeezing with the lips; and it eases response, reducing tension. Weak breath support leads to biting and pinching with the embouchure, and that tension spreads throughout the body.
When you start to feel your embouchure muscles start to tire, allow your face to relax, and focus instead on powerful abdominal breath support.