Running Cocoapods "pod install" Causes Libraries to Disappear - ios

Question

We are having a problem with an existing project. Running pod install causes our existing Pods to break.
Our existing project looks like this:
After I run pod install, several of the libraries that appear in the Products directory and under the Targets are gone, and the app won't compile:
Here is the output from pod install --verbose
myapp-Macbook-Air:myapp username$ pod install --verbose
Preparing
Updating local specs repositories
Updating spec repo `master`
$ /usr/bin/git pull --ff-only
Already up-to-date.
Analyzing dependencies
Inspecting targets to integrate
Using `ARCHS` setting to build architectures of target `Pods`:
(`$(ARCHS_STANDARD)`)
Finding Podfile changes
- Dropbox-iOS-SDK
- FiksuSDK
- GoogleAnalytics
- Localytics
- MobileAppTracker
- SDWebImage
Resolving dependencies of `Podfile`
Comparing resolved specification to the sandbox manifest
- Dropbox-iOS-SDK
- FiksuSDK
- GoogleAnalytics
- Localytics
- MobileAppTracker
- SDWebImage
Downloading dependencies
-> Using Dropbox-iOS-SDK (1.3.13)
-> Using FiksuSDK (4.4.1)
-> Using GoogleAnalytics (3.13.0)
-> Using Localytics (3.5.0)
-> Using MobileAppTracker (3.7)
-> Using SDWebImage (3.7.2)
- Running pre install hooks
Generating Pods project
- Creating Pods project
- Adding source files to Pods project
- Adding frameworks to Pods project
- Adding libraries to Pods project
- Adding resources to Pods project
- Linking headers
- Installing targets
- Installing target `SDWebImage` iOS 7.0
- Installing target `Pods` iOS 7.0
- Running post install hooks
- Writing Xcode project file to `Pods/Pods.xcodeproj`
- Generating deterministic UUIDs
- Writing Lockfile in `Podfile.lock`
- Writing Manifest in `Pods/Manifest.lock`
Integrating client project
Integrating target `Pods` (`myapp.xcodeproj` project)
Adding Build Phase 'Embed Pods Frameworks' to project.
- Running post install hooks
- cocoapods-stats from
`/usr/local/lib/ruby/gems/2.1.0/gems/cocoapods-stats-0.6.2/lib/cocoapods_plugin.rb`
Sending stats
- Dropbox-iOS-SDK, 1.3.13
- FiksuSDK, 4.4.1
- GoogleAnalytics, 3.13.0
- Localytics, 3.5.0
- MobileAppTracker, 3.7
- SDWebImage, 3.7.2
Pod installation complete! There are 6 dependencies from the Podfile and 6
total pods installed.
[!] The `myapp [Debug]` target overrides the `GCC_PREPROCESSOR_DEFINITIONS` build setting defined in `Pods/Target Support Files/Pods/Pods.debug.xcconfig'. This can lead to problems with the CocoaPods installation
- Use the `$(inherited)` flag, or
- Remove the build settings from the target.
(We've always had that GCC_PREPROCESSOR warning)
Any ideas about what be happening here, would be appreciated.
Some other notes: We are having this problem on Xcode 6 and Xcode 7, but both machines are running OS X 10.11, which might be the problem. Cocoapods v0.39.0. Also tried downgrading to Cocoapods v0.25.0 but the same problem occurs.