Instructor Pilot reported a loss of control during landing. The Instructor assumed the controls but was unable to prevent a runway excursion off the right side of the active runway. Brake fluid was found on the left main brake after post flight inspection was performed after aircraft recovery.

Narrative

This event occurred during a normal training sortie with an experienced student. At the end of a 1.8 hour sortie; traffic pattern operations were being performed at ZZZ on Runway XXR. The conditions were turbulent but initially within student training limits. Multiple dust devils were visible within a few miles of the airfield. Winds began to change from light to a strong cross wind fairly quickly. When they finally shifted to 190/10G21; the tower elected to switch to Runway XY. This was when we decided to discontinue training. Due to the turbulent nature of the conditions; I as the instructor took control for the final approach and landing. When we turned base; the tower instructed us to land long. I accepted as this was a normal practice for us and the Runway was XXX0 feet long (much longer than needed to perform a normal landing). I elected to touchdown at approximately 3500 feet remaining as this is a normal Runway length for airfields at which we typically fly. After an uneventful touchdown; I began breaking normally and calling distances and speeds aloud for my student to understand I was paying attention to the rate at which I was slowing. At approximately 800 feet remaining and between 25 and 30 knots; the aircraft immediately began pulling hard to the right. I made every effort to steer back to center line as well as slow the aircraft but was unable to until just off of the Runway. The aircraft came to rest approximately 10 feet from the edge aligned with the Runway. No airport property was damaged; the aircraft was completely intact including the tires and there were no injuries. After exiting the aircraft approximately 1 minute after the event; I noticed a wind sock showing winds from the northwest at greater than 15 knots. It appeared there was a rapid change in the direction of the strong wind during landing roll. The aircraft was able to be towed away normally. Upon inspection of the aircraft approximately 18 hours later there was some brake fluid pooling under the left brake indicating possible issues with that brake but unknown if it was the result of the hard breaking or preexisting. There was no evidence of it prior to flight and the aircraft slowed normally from touchdown until the aircraft began to veer right.