Timestamping using a hashed linked list and public known events
Your diagram is not very clear, but XOR is not a good combiner function to use for timestamping, as it may allow backdating in some circumstances.  For instance, see the "time travel" attacks in Section 3.3 of the following paper (e.g., pp.179-180): Cryptanalytic Attacks on Pseudorandom Number Generators.  Depending upon how many inputs you have to the XOR function and the specific details of your scheme (which are not sufficiently clear from your question), generalized birthday attacks may also be possible and might allow backdating.

Instead, you should use hashing.  There are standard constructions for hash-chaining: I suggest you use them.  In particular, if the sequence of documents/data to be hashed is $D_1,D_2,D_3,\dots$, we form a hash chain by the formula

$$H_i = \text{Hash}(H_{i-1} \,||\, \text{Hash}(D_i)).$$

Then we can publish the $H_i$'s as well as each $\text{Hash}(D_i)$.  See a standard reference on hash chaining and timestamping for more details.



Update (after the question was edited):

No, this is definitely not a good idea, because $x \oplus y$ does not uniquely determine $x$ and $y$.

For instance, someone could register $H_{D_i}$ (which gets mixed with $B_{t_i}$), then later claim that they actually registered $H'_{D_i}$ (and it was mixed with $B'_{t_i}$), as long as $H_{D_i} \oplus B_{t_i} = H'_{D_i} \oplus B'_{t_i}$.  How would you detect it?  There's not a fantastic way to do it.

It would be much better to compute $M_i$ as

$$M_i = \text{Hash}(\text{Hash}(D_i), t_i, B_{t_i}).$$

Now $D_i$ and $B_{t_i}$ are uniquely determined, given $M_i$; the collision-resistance of the hash function prevents backdating or lying about how $M_i$ was formed.  This is such a simple fix, and it is so obviously better, that there's not much point analyzing in great detail the exact degree of vulnerability of xor; it's clear that using a hash is better than using a xor, so there's absolutely no reason to use a xor in this context.

So, bottom line: use a hash function, not xor.