Perhaps no one view been established as a Reformation Arminian position to explain God's enabling grace.
In principle, I think that those who freely claim the appelation Reformation Arminianism. all believe that man cannot, of his own accord, believe in the Gospel. However, I think there are some Arminians who believe that the atonement actually accomplished a victory over the state of human depravity. Thus, they would claim that humanity WAS totally incapable of believing in Jesus UNTIL somehow God's grace in the cross removed it so that everyone may NOW freely respond to the Gospel presentation.
This might be plausible, but I'm inclined to think that man continues to be in a state of Total Depravity which cannot be overcome except by the specific drawing of specific individuals at specific moments by God's grace. If this is the case, then this is point of commonality with the theology of the reformers, in that God does not always draw all men at all times, but only at his own discretion and his own timing.
The caveat is that God generally desires to always draw all men at all times, but implements this general will in conjunction with and at the behest of the prayers and the actions of the saints. Thus, for example, when we pray urgently for someone's salvation, God impels his Spirit to bring conviction and drawing power.
Or, by way of another example, consider a remote people group which theoretically has never had access to the gospel message. In such a case, the sphere of operation for God's convicting grace would be seriously limited (although it is nonetheless operational through natural revelation). However, out of God's general desire to draw all men, God's Spirit might be sent to believers who have some sort of connection with the remote people group, prompting them to pray for the unreached people. Since God is eager to hear such prayers and promises to send more workers into the harvest field, this would increase the sphere of operation for God's convicting grace.
This is both similar to and different from Amyraldianism, if I understand Amyraldianism correctly. The similarity is that God is in control over who is convicted and when he is convicted. The difference is that 1) the Spirit's pre-egenerating grace is intended for all; and 2) his ministry is resistible.