You're asking what a service will do if you send them unconfirmed bitcoins. We can't answer that without knowing their rules and policy.
Edit:
I misunderstood the question. My answer above still stands but most mixer services allows you to generate another address for mixing, you can use that.
Thank you for your help! Would you mind sharing one or more examples with me for mixing services that allow generating new address that works while withdrawal is being processed on the previous one?
Bitmixer.io is one, I use it a lot and never got any problem so far.
[banned mixer] allows you to do this but I only used it once to compare it with bitmixer.io.
Both of them have letter of guarantee. Make sure to download and verify the letter before sending anything.
There are others but I've never used them and they kinda look shady (e.g bitcoinmixer.net looks a lot like bitmixer.io).