Let us keep the notations from above, and let's write $G:=\mathrm{Gal}(E/F)$. Let me quickly recall the origin of the Brauer obstruction: it really comes from the Hochschild-Serre spectral sequence

$$H^p(G,E^q(X_E,\mathbf{G}_m))\Longrightarrow E^{p+q}(X,\mathbf{G}_m)$$

(I'm writing $E^{\ast}=H^{\ast}_{\mathrm{et}}$ for étale cohomology here, because the system doesn't seem to like too many subscripts.) If we analyze this in low degrees, this gives us the following classical exact sequence (for any $F$-variety $X$):

$$0\to H^1(G,E^0(X_E,\mathbf{G}_m))\to\mathrm{Pic}(X)\to H^0(G,\mathrm{Pic}(X_E))\to H^2(G,E^0(X_E,\mathbf{G}_m))\to\ker\left[\mathrm{Br}(X)\to\mathrm{Br}(X_E)\right]$$
$$\to H^1(G,\mathrm{Pic}(X_E))\to H^3(G,E^0(X_E,\mathbf{G}_m))$$

So we'd like to generalize the sequence above to the situation of $\mathrm{CH}^n(X)=H^n(X,\mathcal{K}_n)$ (where $\mathcal{K}_n$ is the Zariski-sheafification of the presheaf $K_n$). Assume $X$ geometrically regular here. The Gersten resolution of $\mathcal{K}_n$ on $X_E$ is the complex

$$C^{\bullet}(X_E)\colon\quad K_nk(X_E)\to\bigoplus_{x\in X_E^1}K_{n-1}k(x)\to\bigoplus_{x\in X_E^2}K_{n-2}k(x)\to\cdots\to\bigoplus_{x\in X_E^{n-1}}K_1k(x)\to\bigoplus_{x\in X_E^n}K_0k(x)$$

There's a similar complex $C^{\bullet}(X)$ giving the Gersten resolution of $\mathcal{K}_n$ on $X$. We regard the complex $C^{\bullet}(X_E)$ as a $G$-complex; write $\sigma$ for the map

$$C^{n-1}(X_E)=\bigoplus_{x\in X_E^{n-1}}k(x)^{\times}=\bigoplus_{x\in X_E^{n-1}}K_1k(x)\to\bigoplus_{x\in X_E^n}K_0k(x)=Z^n(X_E)$$

of $G$-modules. I want to argue that the kernel of this map is playing the role of $E^0(X_E,\mathbf{G}_m)$ for higher $n$. (When $n=1$, this kernel coincides with $E^0(X_E,\mathbf{G}_m)$.)

Now we might hope for a spectral sequence

$$H^p(G,H^q(C^{\bullet}(X_E)[n]))\Longrightarrow H^{p+q}(C^{\bullet}(X)[n])$$

but of course $K$-theory doesn't quite satisfy Galois descent, so we don't have this convergence ($C^{\bullet}(X)[n]$ is not the homotopy fixed-point complex of $C^{\bullet}(X_E)[n]$). But we're only trying to analyze a very small piece of this spectral sequence — the piece involving $\sigma$. For that, Hilbert Theorem 90 does the work, and we get the following exact sequence:

$$0\to H^1(G,\ker\sigma)\to\mathrm{CH}^n(X)\to H^0(G,\mathrm{CH}^n(X_E))\to H^2(G,\ker\sigma)\to\ker\left[H^2\left(G,C^{n-1}(X_E)\right)\to H^2(G,Z^n(X_E))\right]$$
$$\to H^1(G,\mathrm{CH}^n(X_E))\to H^3(G,\ker\sigma)$$

So we find an obstruction to descending cycles of codimension $n$ in $H^2(G,\ker\sigma)$. Is this the sort of thing you had in mind?