The audio from the two resources you mentioned (bolor-toli.com and studymongolian.net) can be downloaded, using these two bookmarklets:
bolor-toli.com:
javascript:!function(){function e(e,t,n,a,r,c){var o="/dictionary/sounds",i={word:a,trans_id:r,lang_id:c,notfound_message:t},l="script",s="POST",u=$.ajax({type:s,url:o,data:i,async:!1,dataType:l,beforeSend:function(e){e.setRequestHeader("X-CSRF-Token",$('meta[name="csrf-token"]').attr("content"))}}).responseText;try{var d=document.createElement("html");d.innerHTML=u;var p=document.createElement("a");p.href=d.querySelector("audio source").src,p.target="_blank",p.innerHTML='<i class="glyphicon glyphicon-floppy-save"></i>',e.parentNode.appendChild(p)}catch(h){}}[].forEach.call(document.querySelectorAll('td a[href="javascript:void(0)"][onclick~="\'2\')"][onclick*="showSounds"]'),function(t){var n=t.getAttribute("onclick");n=n.replace("showSounds(this, ","").replace(")","");var a=n.split("',").map(function(e){return e.trim().substr(1)});a.unshift(t),a[a.length-1]="2",e.apply(null,a)})}();
Use on the search page. Generates a download link next to the audio play button. Note that it takes a while to load, because it has to make a network request to get the link for each audio file. Also, not every word has audio.
studymongolian.net
javascript:!function(){[].forEach.call(document.getElementsByClassName("myButton_play"),function(e){var t=e.getAttribute("onclick").split(",")[2].replace(/'/g,""),n=document.createElement("a");n.target="_blank",n.href=t,n.innerHTML="Download",e.parentNode.appendChild(n)})}();
Generates a "Download" link next to the play button on the page you mentioned.
Of course, I cannot guarantee these two will work, that they will function properly, etc, especially when/if these pages ever change.