An apparatus in one embodiment comprises at least one processing device. The at least one processing device is configured to monitor performance of respective ones of a plurality of paths for accessing a logical storage device, and responsive to detection of at least one specified condition in the monitored performance relating to at least a subset of the paths, to move at least one application from a first container that utilizes a first access protocol to access the logical storage device to a second container that utilizes a second access protocol different than the first access protocol to access the logical storage device. For example, in some embodiments, the at least one processing device is configured to move an application from first container that utilizes a SCSI access protocol to a second container that utilizes an NVMe access protocol, and vice versa, responsive to detected performance issues.