Yes if $L/k$ is smooth. 

$\mathcal H(k)$ is (the homotopy category of) the Bousfield localization of space-valued (simplicial)  sheaves on $Sm/k$ at the maps $X\times \mathbb A^1\to X$ for all $X \in Sm/k$. 

In particular, the basechange $Sm/k\to Sm/L$ induces a functor $Sh(Sm/k)\to Sh(Sm/L)$ on sheaf categories (given by left Kan extension followed by sheafification), which sends representables (so smooth schemes $X/k$) to their basechange (because the representable functor $F = \hom(-,X)$ is mapped to $\hom(-,X_L)$, which is already a sheaf, because the Nisnevich topology is subcanonical), and so it send $X\times \mathbb A^1\to X$ to $X_L\times\mathbb A^1\to X_L$, in particular we get a commutative square of localizations : 

$$\require{AMScd}\begin{CD}Sh(Sm/k) @>>> Sh(Sm/L) \\
@VVV @VVV \\
\mathcal H(k) @>>> \mathcal H(L) \end{CD}$$

So if "down" sends $X\to *$ to an equivalence, so does "down-right", and therefore so does "right-down", but "right" sends it to $X_L\to *$, which proves the claim.