You should read the release notes of the new version of your dependency, fix any obvious issues from that, see if your tests pass, and wait for bug reports to roll in for non-obvious things not caught by automated tests. Ideally you should do this before the new release of the dependency hits the repos of the distros most of your users use, so that it's only the enthusiasts that are hit by unexpected bugs.
Even if you could delay and batch the work together every several releases of a dependency, you're still doing the same amount of work, and it's usually simpler to keep up bit by bit than all at once.
One trick is to not use dependencies that have constant churn and frequent backward-incompatible changes, and to avoid using newly introduced features until it's clear they've stabilised. When you choose dependencies, you're choosing how much work you're signing up for, so choose wisely.
Of course you could go the alternate route and ship all dependencies bundled - but that is a way to ignore technical debt and accidentally end up with a dead project.
Also, your project should not demand a specific version of `libfoo`. If `libfoo` follows semver and a minor release breaks your project, that is a bug in `libfoo`. Deployments of production software should pin versions, but not your project itself.