I am a mega-fan of functional batch programming.
My solution here might be a sloppy mess as I haven't slept NEARLY enough but I can assure you that it works and does more or less what you are asking.
This is possibly the most inefficient way to go about this but if I wrote the batch to do all of the extensions per directory, the keeping track if I told you about them gets a bit more complicated but would be faster execution.
Also, the functions could be reduced to very little code if I didn't use variables so you could follow along as to what was happening.
@echo off
:: Set the starting path that we will be searching.
Set pathToSearch=C:\users\yo_mamma
cd /d "%pathToSearch%"
if not "%ERRORLEVEL%"=="0" echo Path %PathToSearch% not found&&exit /b 1
Set ExtensionsToCheck="*.foo" "*.txt" "*.bar" "*.pdf"
call :EnumerateExtensions %ExtensionsToCheck%
goto :EOF
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:: Function "EnumerateExtensions"
:EnumerateExtensions
if "%~1"=="" goto :EOF
Set fileMatchFound=FALSE
Set searchFileMask=%1
for /r %%p in ('.') do call :EnumerateDirectories "%%p" %searchFileMask%
shift
goto :EnumerateExtensions
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:: Function "EnumerateDirectories"
:EnumerateDirectories
Set directoryPath=%~DP1
Set fileMask=%~2
for %%f in (%fileMask%) do call :MatchFound "%directoryPath%" "%fileMask%" "%%f"
goto :EOF
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:: Function "MatchFound"
:MatchFound
if not "TRUE"=="%fileMatchFound%" echo ****** Mask match found: %~2&&Set fileMatchFound=TRUE
:: echo del %~1%~3 <-- delete the found file here..
goto :EOF