diff --git a/docker/Dockerfile b/docker/Dockerfile
index 7fdd95f..4ff3c28 100644
--- a/docker/Dockerfile
+++ b/docker/Dockerfile
@@ -6,8 +6,8 @@ RUN apt-get update && apt-get install -y curl build-essential && \
     curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | \
     sh -s -- --default-toolchain=nightly --component=miri --component=rust-src --profile=minimal -y && \
     cargo install --git https://github.com/saethlin/miri-tools cargo-download inapty get-args && \
-    cargo install cargo-nextest --locked && \
     cargo install --git https://github.com/RalfJung/cargo-careful --rev 7fa90c6 cargo-careful && \
+    curl -L https://get.nexte.st/latest/linux | tar zxf - && mv cargo-nextest /root/.cargo/bin/ && \
     rm -rf /var/lib/apt/lists/*
 
 COPY nextest.toml /root/.cargo/nextest.toml
diff --git a/docker/run.sh b/docker/run.sh
index 6882024..3f891ca 100644
--- a/docker/run.sh
+++ b/docker/run.sh
@@ -87,4 +87,6 @@ do
         fi
     fi
     echo "-${TEST_END_DELIMITER}-"
+    # Delete everything in our writable mount points
+    find /build /tmp /root/.cargo/registry -mindepth 1 -delete
 done < /dev/stdin