If $X$ is a genus $g\geq 2$ curve over a number field $K$, then $X(K)$ is finite by Falting's Theorem. My question is how does $X_L(L)$ behave for finite field extensions $L/K$? In particular, is there a bound on $X_L(L)$ for $L/K$ that depends only on $X$ and $[L:K]$?
Assuming the conjecture that varieties of general type cannot have a Zariski dense set of points over a number field (which is a consequence of the more precise Vojta conjectures, for example), Patricia Pacelli [1] proved the following, which is much stronger than what you're asking, since the constant only depends on the genus of $X$:
Theorem Fix $d\ge1$ and $g\ge2$. There is a constant $C=C(g,d)$ such that for all number fields $L/\mathbb Q$ of degree at most $d$ and all smooth projective curves $X/L$ of genus at most $g$, one has $\#X(L)\le C(g,d)$.
Pacelli's result is a generalization of an earlier result by Caparaso, Harris, and Mazur [2] that wasn't quite as uniform, but that really set in motion the use of this conjecture to prove uniformity result of this sort.
But if you want unconditional results, I don't think there's anything known in general, although one might(?) be able to get something like $$ \#X(L) \le C(X,[L:K])^{1+\text{rank Jac}_X(L)}. $$
[1] Patricia L. Pacelli, MR 1448017 Uniform boundedness for rational points, Duke Math. J. 88 (1997), no. 1, 77102.
[2] Lucia Caporaso, Joe Harris, and Barry Mazur, MR 1325796 Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), no. 1, 135.

$\begingroup$ Thanks! I am indeed interested in unconditional results. Just to be clear, the rank of the Lpoints of the Jacobian of X is not known to be bounded by $[L:K]$? Is it flat out false? $\endgroup$ – Andrew NC Oct 19 '16 at 3:26

$\begingroup$ Sorry, I meant bounded by a constant that depends only on $[L:K]$ and $X$. $\endgroup$ – Andrew NC Oct 19 '16 at 3:27

$\begingroup$ @AndrewNC The rank is bounded by a constant depending on $X$ and $[L:K]$ and the size of the class group of $L$ (or even on, say, $p$ and the $p$rank of the class group of $L$ for any fixed prime $p$), but that class rank is not known to be bounded by $[L:K]$. There are some current conjectures about uniform boundedness of ranks of abelian varieties, but they are still just conjectures, and somewhat controversial at that. $\endgroup$ – Joe Silverman Oct 19 '16 at 9:56