It's complicated; I suggest asking your GM
Both booming blade spells are still around, just not active
The Player's Handbook has a section "Combining Magical Effects" which states:
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.
We can establish from this that if somebody affected by multiple instances of booming blade, both instances are still around, just not necessarily active. We also see that only the most potent of the castings has any effects at a given time.
Is damage an effect of a spell? I personally say that it is.
This is an important question because if damage is not an effect of a spell, then simultaneous instances of damage can occur since nothing is preventing them from overlapping.
I address this question somewhat in my own answer to the following question:
To reiterate here though, one could certainly argue that damage is not an effect of a spell (people have done exactly that), however I personally believe that damage is an effect of the spell. I base some of this on the Dungeon Master's Guide errata (direct download) which added the "Combining Game Effects" section (emphasis mine):
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. [...]
The Fire Elemental's Fire Form trait states:
[...] The first time it enters a creature's space on a turn, that creature takes 5 (1d10) fire damage and catches fire; until someone takes an action to douse the fire, the creature takes 5 (1d10) fire damage at the start of each of its turns.
Here we see damage that just happens at the start of one's turn and if multiple instances of this damaging effect are on a creature, only one occurs. The damage does not stack, which leads me to believe that damage is an effect of a spell, and so the damage from both castings of booming blade would not both occur at once.
Do both instances of booming blade even activate when you move? Ask your GM.
The following question exists and provides some insight into this question:
The current answers there both state the target in question only needs to make one saving throw because only the effects of one instance of the spell will be active at any given moment. They also state that should one of the castings end, the other will remain and its effects will become active.
Unfortunately that doesn't address the issue here since booming blade states:
[...] If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends. [...]
Here's the problem, is the spell ending an effect of the spell? In other words, does the inactive casting also end when the target moves? Can that second casting even register or respond to the fact that you moved given that it is inactive? I honestly have no idea and have asked this as its own question. That said, as far as I'm aware, there are two possibilities:
Both instances end at the same time
The target will take either 1d8 if damage is an effect of a spell and 2d8 if damage is not an effect of a spell and both spells will end.
Only one instance ends at a time
The target will take 1d8 damage and that casting will end meaning the target will still have the other casting around and that casting will now become active. Thus if the target moves again they will take another 1d8 damage from the second casting which will then end.
In summary, the target either takes 2d8 damage at once and both spells ends, takes 1d8 damage and both spells end, or takes 1d8 damage and only one spell ends and they will take another 1d8 damage should they continue moving further.