Obviously, when manually beatmatching (think vinyl! LOL) the only option is keeping your finger on the label and speeding up/slowing down the disc. One of the reasons that pre-90s vinyl DJs knew a LOT of transitions that didn’t need lengthy transitions.
Software like Serato and Mixvibes Cross have a pretty good beatgridding engine that can work with “hand-drummed” material. Also they allow to set stuff straight (time-consuming as it is, it beats audacity for the purpose imho).
Another option if you have availability of Ableton Live, is to use that. Here too it’s easier to work with (warp) non-mechanical beats.
Hope that helps a bit. Sure others will pitch in with more suggestions and tips.