I have a doubt whether the entities in pillager patrols despawn or not. If they do, after how much time do they despawn? I am asking this because one patrol has been following me forever and they do not seem to despawn. My version is Java edition 1.16.5. I have googled it and cannot seem to find an answer.
After some research, I could find no documented special despawn rules for pillager patrols. According to despawn mechanics, any mob 32 or more blocks away from the player after 30 seconds has a 2.5% chance per second to despawn at random.
What may have happened is that you moved far enough away before the despawn time could occur that the chunk had become unloaded. Mobs in an unloaded chunk do not despawn and are saved along with everything else within it.
If you are adamant on not fighting a patrol because you are ill-equipped, you should put yourself at a great distance just far enough that you can still see them but are definitely more than 32 blocks away and wait for a while.
If you are adamant on not fighting a patrol because you do not want to suffer the effects of "Bad Omen" you can avoid getting the effect by either drinking milk after receiving the effect or by killing the banner-carrying pillager in a way that will not be attributed to you, such as dumping a bucket of lava underneath their feet.
An alternate way of looking at this follows: The other night I was in the desert smacking husk walkers around in large numbers for my rotten flesh collection. The Cleric has emeralds! I rounded a sand dune, and walked straight into a freshly spawned patrol 30 blocks away out in the open. Yikes! facing 4 crossbows at once in plain iron armor seemed foolhardy, so I scuttled back behind the edge of the dune. Steeling myself with my enchanted diamond sword, I skirted the dune closely moving forward once again, and the patrol had vanished. Catching a glimpse of one Illager part way up the dune, I downed him by striking his legs repeatedly from below, while taking not a single crossbow bolt. I climbed the dune searching the vast open desert under moonlight.
Nothing. The rest of the patrol had apparently been swallowed up by the vast sands without a trace (despawned?)
I was disappointed! You see, receiving the Bad Omen will trigger an attack on the next village you wander into. After you defeat a number of waves, you will pick up the Totem of Undying from the Warlock at the end. It gives you an in-place respawn with all of your items intact if you hold it in you off hand. Don't pass that up...