To continue the trend of "continuously modifying entity data", this time, when continuously empty the body armor of horse, llama or trader llama or empty the hand of allay using "/data merge", then right click to equip the horse armor/carpet of horse/(trader) llama or right click the allay to hold an item, it looks like it's wearing horse armor or carpet or holding an item, but when press E on the horse or the (trader) llama, or "Save and quit" then enter the world, or unload then load the chunks, the horse and the (trader) llama no longer wear the horse armor or the carpet and the allay no longer holds an item.

STR:
Step 1: Summon a horse, a llama, a trader llama or an allay.
Step 2: Place down an repeating command block, right click on the command block, then type:

"/execute as @e[type=horse/llama/trader_llama] run data merge entity @s {body_armor_item:{id:"minecraft:air"}}" or "/execute as @e[type=allay] run data merge entity @s {HandItems:[

,{}]}"

, then set to "Always Active" and then click "Done".
Step 3: Right click on the horse while holding a horse armor, on the llama or trader llama while holding a carpet or on the allay while holding any item -> The horse armor/carpet/item visually here.
Step 4: Press E while riding the horse, llama or trader llama, or "Save and Quit" then rejoin the world -> The horse armor/carpet/item is gone.