Mail.app extremely slow on OS X 10.9 Mavericks
Thanks for uploading the Logfile. There are several entries which could describe the Problems, f.e.:

25/10/13 4:29:58.408 pm WindowServer[125]   disable_update_timeout: UI updates were forcibly disabled by application "Mail" for over 1.00 seconds. Server has re-enabled them.
25/10/13 4:30:12.256 pm WindowServer[125]   disable_update_likely_unbalanced: UI updates still disabled by application "Mail" after 15.00 seconds (server forcibly re-enabled them after 1.00 seconds). Likely an unbalanced disableUpdate call.


and

25/10/13 4:36:30.089 pm WindowServer[125]   common_reenable_update: UI updates were finally reenabled by application "Mail" after 392.86 seconds (server forcibly re-enabled them after 1.00 seconds)
25/10/13 4:37:26.568 pm com.apple.IconServicesAgent[437]    main Failed to composit image for binding VariantBinding [0x159] flags: 0x8 binding: FileInfoBinding [0x359] - extension: jpg, UTI: public.jpeg, fileType: ????.


and 

25/10/13 4:45:13.084 pm Mail[408]   xpc __securityd_create_connection_block_invoke got event: Connection interrupted


and

25/10/13 4:45:21.796 pm com.apple.internetaccounts[463] An instance 0x7fcb64cef770 of class IMAPMailbox was deallocated while key value observers were still registered with it. Observation info was leaked, and may even become mistakenly attached to some other object. Set a breakpoint on NSKVODeallocateBreak to stop here in the debugger. Here's the current observation info:
&lt;NSKeyValueObservationInfo 0x7fcb64cf0620&gt; (
&lt;NSKeyValueObservance 0x7fcb64cf0750: Observer: 0x7fcb64f28160, Key path: uidNext, Options: &lt;New: NO, Old: NO, Prior: NO&gt; Context: 0x12131544b, Property: 0x7fcb64cf05f0&gt;
)


Some of the entries describe Timeouts because of syncing errors. How to fix them? 

I would cleanup the Cache of the Mailaccount either by moving the Cache Folder or be deleting and recreating the IMAP-Account.

If you want to cleanup the Cache you have to:


Quit Mail
got to ~/Library/Mail/V2
Find the Folder for your IMAP-Account in the Folder "V2". It should be named something like "IMAP-you@email.address"
move the Folder for your IMAP-Account somewhere else, f.e. to the Desktop
start Mail again. It should be syncing the Mails from your IMAP-Account to your Computer in a normal fashion.


If that does not help you could try the second approach: deleting the IMAP-Account and recreating it. 

Try that and post your results please.